AD/ A - 002 320
DIGITAL FLIGHT CONTROL SYSTEM FOR TACTICAL FIGHTER. VOLUME I. DIGITAL FLIGHT CONTROL SYSTEM ANALYSIS
A. Ferit Konar, et al
Honeywell, Incorporated
Prepared for:
Air Force Flight Dynamics Laboratory
June 1974
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National Technical Information Service U. S. DEPARTMENT OF COMMERCE
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This technical report has been reviewed and is approved for publication.
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Chief, Control Systems Development Branch
Flight Control Division
Air Force Flight Dynamics Laboratory
Copies of this report should not be returned unless return is required by security considerations, contractual obligations, or notice on a specific document.
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REPORT DOCUMENTATION PAGE |
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1. REPORT NUMBER |j GOVT ACCESSION NO. AFFDL-TR-73-119 , |
J RECIPIENT'S CATALOG NUMBER |
A TITLE (md Mini*) DIGITAL FLIGHT CONTROL SYSTEM FOR TACTICAL FIGHTERS, Volume I. Digital Flight Control System Analysis |
5 f YPE 0 ^ REPORT S PERIOD COVERED INTERIM REPORT Feb. 1972 through June 1973 |
« PERFORMING ORG. REPORT NUMBER F-0131-IR1. 08001 |
|
y author r«; A, Ferit Konar J. K. Mahesh B. Kizilos |
e contract or grant number^; F33615-72-C-1058 |
• PERFORMING ORGANIZATION NAME AND AOORFSS Honeywell Inc. , Systems and Research Center, 2700 Ridgway Parkway N. E. , Minneapolis, Mn. 55413 |
to PROGRAM ELEMENT. PROJECT. TASK AREA A WORK UNIT NUMBERS Project 1987, Task 198701 |
II CONTROLLING OFFICE NAME AND ADDRESS U. S. Air Force Flight Dynamics Laboratory |
17 REPORT DATE June 1974 |
Wright Patterson Air Force Base, Ohio 45433 |
n NUMBER OF PAGES |
14 MONITORING AGENCY NAME • AOORESS/if dillntni It nm Controlling Olhrt) |
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If KEY WOPOS (Continue on rrvtrif aid* it neceetary and identity bv block number) Flight Control Digital Tactical Fighter |
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10 ABSTRACT (Continue on rover ee aide It neceeeary and Identity by block number) The Digital Flight Control Systems for Tactical Fighters Program is a development program which defines the technology necessary to apply digital flight control techniques to the three -axis, multiple flight control configura- tion demands of advanced fighter aircraft. Analysis efforts to date have defined powerful computer program tools which permit determination of flight control system performance as a function of computational parameters -- word length, sample rate, and computational delays. An exercise of the |
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DD 1 JAN *71 1473 EDITION OF 1 NOV SS IS OBSOLETE |
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programs using the F-4 as a model indicates 100 iterations per second as satisfactory for the longitudinal axis. ..
Requirements for digital flight control systems have been defined in all hardware/software development areas including the following:
• Sizing rules for all types of digital flight control functions. Control law sizing rules are nearly independent of programming form.
I • Impact of redundancy and self-test on computer requirements. Self- test techniques to greater than 97 percent effectiveness are feasible.
• Input and output signal interface complement including recommended i multiplexing techniques.
• EMI provisions which must be implemented in the initial system design.
In general, the applications of the requirements are noted to be still dependent upon the relative importance of reliability, maintainability, cost, and physical constraints.
Integration studies have shown the feasibility of including outer loop and multimode functions in the digital flight control system configuration. Multi- rate structures and switching strategies are required for this integration.
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FOREWORD
This report was prepared by Honeywell Inc., Minneapolis, Minnesota 55413, under Air Force Contract F33615-72-C-1058 "Digital Flight Control Sys- tems for Tactical Fighters. " It was initiated under Project No. 1987, "Manual and Automatic Control Systems Technology," Task No. 198701, "Mission-Oriented Control Laws and Mechanizations. " The work was ad- ministered under the direction of the Air Force Flight Dynamics Laboratory, Wright Patterson Air Force Base, Ohio 45433 by Captain K. W. Bassett, AFFDL/FGL, Project Monitor.
This study is reported in two parts: The Interim Report and the Final Report.
The Interim Report consists of three volumes as follows:
Volume I. Digital Flight Control Systems Analysis
Volume IT. Documentation of the Digital Control Analysis Software (DIGIKON)
Volume III. Digital Flight Control System Design Considerations
This is Volume I of the Interim Report. It covers work performed between February 1972 and June 1973. The contractor's report number is Honeywell Document F-0131-IR 1. ,
The technical work reported in this voMime was conducted by the Research Department of the Systems and Research Division of Honeywell Inc.
Dr. A. Ferit Konar was the principal investigator. Messrs. J. K. Mahesh, Mike Ward and Victor Falkner w ejf the programmer analysts, Mrs. Betty Kizilos and Miss Jane Gayl and Mrs. Marion Borow were the programmers. Dr. E. E. Yore and Mr. D. R.Splschlaeger were the Program Managers. Technical consultation was p^Kded by D. L. Markusen, Dr. Peter Tna0nfelB.
ihe investigators in thisi his guidance and suppoi^ Capt. Vince Darcy for M programs.
■y would like to thank Capt. K. W. Bassett for phis program. They would also like to thank fading direction and assistance in testing the analysis
SECTION I INTRODUCTION 1
SECTION II ANALYSIS APPROACH 3
Stability and Performance Analysis Program 3
Control Law Generation (Synthesis) 6
Digitization Versus Direct Digital Design 6
Computational Requirements and Parametric Study 10 Parametric Study of "Structural Filter" in F-4 11
Longitudinal Control System
Parametric Study of F-4 Longitudinal Control 11
System
MODELING FOR THE DYNAMICS OF MULTIRATE MULTILOOP SYSTEMS
Development of the Linear System Matrices from the Simulation Equations
Implementation of the Simulation Equations / Modeling With Transfer Function Input Transfer Function and Its Quadruple Overall System Quadruple Transformations in State Space Discrete Matrix Model for the Physical Plant Selection of Transition Time Automatic Exponentiation and its Relation to Direct Digital Design
Discrete Matrix Model for the Digital Controller Steady State Gain Prewarping for Pole Placement State Model of the Discrete System in the w-Plane Example
Overall System Modeling for Single Rate Systems Algebraic Controller Dynamic Controller Parametric Interconnection Model and Interconnection Quadruple Overall System Modeling foi#Multi-Rate Systems Delay System Modeling Jf Gust Response Modeling ire Sample Time Effects Discrete System Modeling by Software - Discrete Single-Rate System
Discrete System Modeling by Software - Multi - Variable Multirate System Modeling with Computational Delays
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SECTION IV
SECTION V
TAHLE OK CONTENTS -- CONTINUED
Total Transition Approach Incremental Transition Approach Software Implementation Example of Two-Rate Modeling by Software Mathematical Modeling for Word Length Effects Data Truncation Model Digital Controller Scaling Model Form of Scaling Scaling Constraints Digital Controller Noise Model
SYSTEM PERFORMANCE MODELING IN STATE SPACE
Modeling for Poles and Zeros (POZK)
Response Modeling for Real and Complex Inputs Development of Complex Response Model Complex Response Model for Multirate Systems
The General Frequency Response Software (FREQK)
Pseudo-Zeros of an Output/Input Pair Demonstration Example for FREQK RMS Response Model for Systems with Continuous and Digital Noise Inputs (COVK)
RMS Response of Plant to Continuous Stationary Inputs
RMS Response Model for Digital Controllers with Discrete Inputs (Roundoff Noise)
RMS Response Model for Overall System Gust Response Ratio Word-Length Roundoff Noise Relations Power and Power Spectral Density Modeling for Frequency Truncation (POWK)
Time Response Model for Deterministic Inputs (TRESPK)
COMPUTATIONAL REQUIREMENTS AND PARAMETRIC STUDY
Parametric Study of a Structural Filter in the F-4 Longitudinal Control System Power Content Analysis with Structural Filter Parametric Study of F-4 Longitudinal Control System Modeling of F-4 Longitudinal Control System
Page
69
70 72 74 81 81 82 82 84 87
90
90
92
93
95
96
99
101
107
107
110
111
113
119
121
124
125
126
172
178
178
vi
TABLE Ol-' CONTENTS -- CONCLUDED
Page
Stability and Frequency Response Performance 214 Gust Response Ratio Performance 252
Effects of Computational Time Delay on 261
Longitudinal Control System Stability
SECTION VI CONCLUSIONS AND RECOMMENDATIONS 280
Significant Results 280
Recommendations for Future Analysis Work 281
Recommendations for Future Software Develop- 2 81
ment Work
Conclusions 282
REFERENCES 283
APPENDIX A SIMKTC — MODEL FOR F-4 CONTROLLER VIA TRANSFER FUNCTION INPUT
APPENDIX B STATE MATRIX APPROACH TO ROUND-OFF NOISE ANALYSIS OF DIGITAL FILTERS
APPENDIX C MODELING OF F-4 LONGITUDINAL CONTROL SYSTEM WITH TIME DELAY
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LIST OF ILLUSTRATIONS
Figure Page
1 DIGIKON Software Program for Sample Rate /Word Length 4
Determination
2 Basic Structure of the Digital Control Software System 5
(DIGIKON)
3 Control and Computational Requirements 7
4 Interactive Analysis, Design, and Performance Evaluation 7
in the s- Plane
5 Interactive Analysis, Design, and Performance Evaluation 8
in the s-z Plane
6 Interactive Analysis, Design, and Performance Evaluation 9
in the z-w Plane
7 F-4 Simulation Interconnection 12
8 Block Diagram of a Typical Longitudinal Channel 15
9 The Simulation Matrix 17
10 Simulation Diagram of the Short Period Dynamics 19
1 1 Longitudinal Controller 20
12 Longitudinal Controller Simulation Diagram 20
13 Overall Simulation Diagram 21
14 Input Frobenius Form State Diagram of a Single Input, 24
Single Output Transfer Function
15 Block Diagram of a System Containing Three Transfer Blocks 25
16 State Diagram of Physical Plant Including Hold Elements 28
17 State Diagram of the Digitized Controller 36
18 State Diagram of the Discrete System in the w- Plane 43
19 State Diagram of HQ(z) 45
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LIST OF ILLUSTRATIONS -- CONTINUED
Figure Page
20 Transformation of /-Plane to w- Plane 48
21 Transformation of w - Plane to z -Plane 50
22 Block Diagram of a Single Sample Rate System 52
23 Parametric Interconnection Model 55
24 Two-Rate Algebraic Control System 57
25 Computational Delay Model 58
26 Timing Program for Delay System 58
27 Gust Response Model for Sample Time Effects 61
28 Simple Digital Control System with Continuous Disturbance 63 Input
29 Periodic Variance Response 65
30 Block Diagram of a Single Rate System 66
31 Time Behavior of State Transitions 67
32 Flow in STAMK for Multirate System Modeling with 72
Computational Delays
33 System Block Diagram 73
34 Discrete System Timing Program 74
35 Flow Chart of Subroutine HSIMK for a Two-Rate System 75
with Computational Delays
36 System Block Diagram 76
37 Updating Sequence During One Program Cycle 77
38 HSIMK Flow Chart 79
39 Model by Software for K = 0. 5, T = 1 second 80
40 Control Law with a Single Scale Factor 84
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l /IST OK ILI illSTHATIONS -- CONTINUED
Figure
Scaled Control Equations
Arithmetic Response Matrix Time History
Block Diagram for Arithmetic Response Matrix Generation
Bounds of Arithmetic Response r..(t)
Noise Model for One Arithmetic Cell Multiple Input Samples in a Program Period Frequency Response Evaluations
Block Diagram of a System for an Output-Input Pair r^, u^
Quadruple Input Program
Quadruple Input Image
Poles and Zeros
Plot of db versus uw
Plot of Phase versus uw
Plant Block
Input Functions to Plant
Roundoff Noise Model for the Controller
Overall System RMS Response Model
Continuous Control System
Sampled -Data Control System
State Diagram of the Sampled-Data System
Design Procedures Trade
Roundoff Noise Model of the Digitized Controller Dynamics
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LIST OF ILLUSTRATIONS -- CONTINUED
Figure |
Page |
|
63 |
Stochastic Inputs to a Control System |
122 |
64 |
Power Spectral Density and Power as a Function of u |
123 |
65 |
Replacement of Continuous Controller with a Digital Controller in a Feedback System |
127 |
66 |
Notch Filter Simulation Diagram arvi Equations |
128 |
67 |
Notch Filter Simulation Program Listing |
129 |
68 |
Notch Filter Sample -Time Root Locus in the s- Plane |
130 |
69 |
Notch Filter Sample-Time Root Locus in the z-Plane |
131 |
70 |
Filter Quadruple and Associated Poles and Zeros for Sample Time T = 0 sec and Word Length = Full Bits |
132 |
71 |
Filter Quadruple and Associated Poles and Zeros for Sample Time T = 1/1000 sec and Word Length = 16 Bits |
133 |
72 |
Filter Quadruple and Associated Poles and Zeros for Sample Time T = 1/160 sec and Word Length = 16 Bits |
134 |
73 |
Filter Quadruple and Associated Poles and Zeros for Sample Time T = 1/80 sec and Word Length = 16 Bits |
135 |
74 |
Filter Quadruple and /ssociated Poles and Zeros for Sample Time T = 1/40 sec and Word Length =16 Bits |
136 |
75 |
Filter Quadruple and Associated Poles and Zeros for Sample Time T = 1/20 sec and Word Length =16 Bits |
137 |
76 |
Effect of Coefficient Word Length on the Notch Filter Quadruple (Sample Time = 1/80 sec) |
138 |
77 |
Filter Gain (db) versus Omega for Sample Time T = 0 sec and Word Length = Full Bits |
139 |
78 |
Filter Phase (deg) versus Omega for Sample Time T = 0 sec and Word Length = FuF Bits |
141 |
79 |
Filter Gain (db) versus Omega for Sample Time T = 1/160 |
143 |
sec and Word Length =16 Bits
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LIST OF ILLUSTRATIONS- -CONTINUED
figure |
Page |
|
80 |
Filter Phase (deg) versus Omega for Sample Time T = 1/160 sec and Word Length = 16 Bits |
145 |
81 |
Filter Gain (db) versus Omega for Sample Time T = 1/80 sec and Word Length = 16 Bits |
147 |
82 |
Filter Phase (deg) versus Omega for Sample Time T = 1/80 sec and Word Length = 16 Bits |
149 |
83 |
Filter Gain (db) versus Omega for Sample Time T = 1/40 sec and Word Length = 16 Bits |
151 |
84 |
Filter Phase (deg) versus Omega for Sample Time T = 1/40 sec and Word Length = 16 Bits |
153 |
85 |
Filter Gain (db) versus Omega for Sample Time T = 1/20 sec and Word Length = 16 Bits |
155 |
86 |
Filter Phase (deg) versus Omega for Sample Time T = 1/20 sec and Word Length = 16 Bits |
157 |
87 |
Filter Gain (db) versus Omega for Word Length = 24 Bits and Sample Time T = 1/1000 sec |
159 |
88 |
Filter Gain (db) versus Omega for Word Length =16 Bits and Sample Time T = 1/1000 sec |
161 |
89 |
Filter Gain (db) versus Omega for Word Length = 12 Bits and Sample Time T = 1/1000 sec |
163 |
90 |
Filter Gain (db) versus Omega for Word Length = 8 Bits and Sample Time T = 1/1000 sec |
165 |
91 |
Frequency Response Table, Omega versus Sample Time |
167 |
92 |
Loss of Phase Margin versus Sample Time at Omega = 10. 25 rad/sec |
171 |
93 |
Power Content Analysis Model |
172 |
94 |
System Quadruple and Outputs |
174 |
95 |
Power and PSD (200 rad/sec Input Filter) |
175 |
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LIST OF ILLUSTRATIONS --CONTINUED
Figure |
Page |
|
96 |
Plots of Power and PSD (200 rad /sec Input Filter) |
176 |
97 |
Plots of Power and PSD (0. 2 rad/sec Input Filter) |
177 |
98 |
F-4 Simulation Interconnection |
180 |
99 |
Sensor Block Diagrams |
181 |
100 |
Sensor State Diagrams |
181 |
101 |
Sensor Equations |
182 |
102 |
Program Listing for Sensors |
183 |
103 |
Longitudinal Aeroelastic Equations of Motion |
184 |
104 |
Vehicle Simulation Diagram |
185 |
105 |
Vehicle Equations |
186 |
106 |
Flight Condition Data |
188 |
107 |
Program Listing for Vehicle Equations |
192 |
108 |
Actuator Block Diagram |
193 |
109 |
Actuator State Diagram |
194 |
110 |
Actuator Equations |
195 |
111 |
Program Listing for Actuator Equations |
197 |
112 |
Controller Block Diagram |
198 |
113 |
Controller State Diagram |
199 |
114 |
Controller Equations |
200 |
115 |
Program Listing for Controller Equations |
202 |
116 |
Plant Block Diagram |
203 |
117 |
Plant Equations |
204 |
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LIST OF ILLUSTRATIONS- -CONTINUED
Figure Page
118 Program Listing for Plant Equations 206
119 Overall System Block Diagram 209
120 Overall System Equations 210
121 Program Listing for Overall Systems Equations 212
122 Sapiple Time Root Locus in the Image s-Plane (Closed 216
Loop F-4)
123 Sample Time Root Locus in the Image z-Plane (Closed 217
Loop F-4 Vehicle Modes)
124 Closed Loop Overall System Poles (T = 0 sec) 218
125 Closed Loop Overall System Poles (T = 1/1000 sec) 219
126 Closed Loop Overall System Poles (T = 1/160 sec) 220
127 Closed Loop Overall System Poles (T = 1/80 sec) 221
128 Closed Loop Overall System Poles (T = 1/40 sec) 222
129 Closed Loop Overall System Poles (T = 1/20 Sec) 223
130 F-4 Control System Open Loop Frequency Response 226
(T = 0 sec)
131 F-4 Control System Open Loop Frequency Response 226
(T = 1/160 sec)
132 F-4 Control System Open Loop Frequency Response 227
(T = 1/80 sec)
133 F-4 Control System Open Loop Frequency Response 228
(T = 1/40 sec)
134 F-4 Control System Open Loop Frequency Table 229
135 Open Loop F-4 Gain (db) versus Omega Plot (T = 0 sec) 233
136 Open Loop F-4 Phase (deg) versus Omega Plot 235
(T = 0 sec)
xiv
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LIST OF ILLUSTRATIONS- -CONTINUED
Figure |
Page |
|
137 |
Open Loop F-4 Gain (db) versus Omega Plot (T = 1/160 sec) |
237 |
138 |
Ooen Loop F-4 Phase (deg) versus Omega Plot (T= 1/160 sec) |
239 |
139 |
Open Loop F-4 Gain (db) versus Omega Plot (T = 1/80 sec) |
241 |
140 |
Open Loop F-4 Phase (deg) versus Omega Plot (T = 1/80 sec) |
243 |
141 |
Open Loop F-4 Gain (db) versus Omega Plot (T = 1/40 sec) |
245 |
142 |
Open Loop F-4 Phase (deg) versus Omega Plot (T = 1/40 sec) |
247 |
142A |
Open Loop Gain (db) versus Omega Plot for Different Sample Times |
249 |
142B |
Open Loop Phase (deg) versus Omega Plot for Dif.\irent Sample Times |
250 |
143 |
Name List for the Overall System States, Inputs, and Outputs |
253 |
144 |
Closed Loop Vehicle Gust Variance Response Ratio versus Sample Time |
255 |
145 |
Variance Response Ratios for Sample Time T = 1/1000 sec |
256 |
146 |
Variance Response Ratios for Sample Time T = 1/160 sec |
257 |
147 |
Variance Response Ratios for Sample Time T = 1/80 sec |
258 |
148 |
Variance Response Ratios for Sample Time T = 1/40 sec |
259 |
149 |
Variance Response Ratios for Sample Time T = 1/20 sec |
260 |
150 |
Time Delay Root Locus in the Z-plane for Sample Time T = 1/40 sec (Overall Closed Loop) |
262 |
151 |
Time Delay Root Locus in the Image s- Plane for Sample Time T = 1/40 sec (Overall Closed Loop) |
263 |
152 |
Closed Loop Overall System Quadruple Using HSIMK for Sample Time T = 1/40 sec, and Time Delay Td = 0 |
264 |
xv
LIST OF ILLUSTRATIONS- -CONCLUDED \
’igure |
Page |
|
53 |
Closed Loop Overall System Quadruple Using HSIMK for Sample Time T = 1/40 sec, and Time Delay T^ = T/4 |
266 |
54 |
Closed Loop Overall System Quadruple Using HSIMK for Sample Time T = 1/40 sec, and Time Delay T^ = T/2 |
268 |
55 |
Closed Loop Overall System Quadruple Using HSIMK for Sample Time T = 1/40 sec, and Time Delay T^ = T |
270 ] |
56 |
Closed Loop Overall System Quadruple Using SIMK for Sample Time T = 1/40 sec, and Time Delay T^ = 0 |
273 j |
57 |
Poles of the Overall Closed Loop System for Sample Time T = 1/40 sec and Time Delay T^ = 0 |
276 |
58 |
Poles of the Overall Closed Loop System for Sample Time T = 1/40 sec and Time Delay T^ = T/4 |
277 |
59 |
Poles of the Overall Closed Loop System for Sample Time T = 1/40 sec and Time Delay Td = T/2 |
278 |
60 |
Poles of the Overall Closed Loop System for Sample Time T = 1/40 sec and Time Delay T , = T |
279 |
LIST OF TABLES
Table |
Page |
|
1 |
s-Plane to z-Plane Transformation |
44 |
2 |
z-Plane to w-Plane Transformation |
44 |
3 |
w-Plane to z-Plane Transformation |
44 |
4 |
Sequence of Transitions |
59 |
5 |
Forms of Transitions of Dynamical Subsystems |
68 |
6 |
Discrete System State Update Sequence |
73 |
7 |
Plant and Controller Data |
76 |
Discrete System Update Sequence 77
5 " t) Variables for Various Frequency Responses 98
Plant and Controller Data as a Function of Sample Time T 116 Parameter Values 172
Phase Margin versus Sample Time (First Crossing) 224
Gain Margin versus Sample Time (First Crossing) 224
Gain Margin versus Sample Time (Second Crossing) 224
Gain Margin versus Sample Time (Third Crossing) 225
u
Frequency Ratios p = — !25L 251
ui
Gust Response Variance Response Ratio (in Percent) as 254
Function of Sample Times (Closed Loop ± Vehicle Variables)
Sufficient Sample Rate Requirements versus Bandwidth 2 81
SECTION I INTRODUCTION
The general objective of this program is to develop the technology necessary to apply digital flight control techniques to the three-axis, multiple flight control configuration demands of advanced fighter aircraft. Specifically, the techniques and requirements of digital flight control systems are to be estab- lished, and a simulation employing a proven airborne digital computer is to be used to validate these requirements.
The Interim Report consists of three volumes as follows:
VOLUME I -- DIGITAL FLIGHT CONTROL SYSTEMS ANALYSIS
VOLUME II — DOCUMENTATION OF DIGITAL CONTROL ANALYSIS PROGRAMS (DIGIKON)
VOLUME III -- DIGITAL FLIGHT CONTROL SYSTEM DESIGN CONSIDERATIONS
This document reports the analytical developments on the Digital Flight Control Systems Analysis which pertain to the specific objective of defining computational requirements for a tactical fighter and determining its per- formance sensitivity to digital flight control system (DFCS) parameters.
Section II presents the analysis approach. The stability and performance analysis program is briefly reviewed. Subsequently, for background infor- mation, the process of generating complete digital control laws is given, and a parametric study of the F-4 longitudinal system is described.
In Section III the technique for mathematical modeling of the computer- controlled system in state space is presented. This is an automated process which has been applied to multivariable, multirate systems. Effects of computational delays are included.
Modeling of performance in state space is described in Section IV. Five performance measures are considered: (1) poles-zeros, (2) frequency response, (3) RMS response to turbulence and roundoff noise, (4) power- content analysis, and (5) time response.
The computer programs which implement the mathematical analyses and models presented in this volume are documented in Volume II.
Some of the analytical developments reported in Volume I have not been incorporated into the existing software due to lack of resources.
A demonstration example is given in Section V to illustrate how these pro- grams are used and how the computational requirements are derived.
1
In Volume III, Digital Flight Control System Design Considerations, the requirements for converting a general digital flight control system to actual hardware and software are addressed. Topics such as sizing rules, input' output information flow, multiplexing, redundancy and self-test techniques and electromagnetic interference requirements are discussed. The results include guidelines to aid in the estimation of the complexity of the actual hard- ware design for both dedicated and general-purpose processor configurations.
In additon. Volume III considers the impact of DFCS design requirements of two practical design applications. The first of these concerns the integration of outer- loop flight control modes with inner- loop functions. The second considers the implementation of multimode control functions. A primary investigation in the second application involved switching strategies to mini- mize transients when changing modes.
The Final Report, AFFDL-TR-74-69, documents the work done in a continua- tion of this study on digital flight control requirements. This effort involved validation of the analysis techniques and applicable design considerations des- cribed in this report. The validation was accomplished through a parameter variation analysis on the F-4 longitudinal axis using the analysis tools of Volumes I and II, and through a simulation test program using a digital airborne computer containing F-4 control functions. A condensation of the analysis and design requirements documented in the three volumes of this report is also included in the final report.
NOTICK
When Government drawings, specifications, or other data are used for any purpose other than in connection with a definitely related Government procure- ment operation, the United States thereby incurs no responsibility nor any obligation whatsoever; and the fact that the government may have formulated, furnished, or in any way supplied the said drawings, specifications, or other data, is not to be regarded by implication or otherwise as in any manner licensing the holder or any other person or corporation, or conveying any rights or permission to manufacture, use, or sell any patented invention that may in any way be related thereto.
This technical report has been reviewed and is approved for publication.
I
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AFFDL-TR-73-119 VOLUME I
i
DIGITAL FLIGHT CONTROL SYSTEMS FOR TACTICAL FIGHTERS
Volume I: Digital Flight Control Systems Analysis
HONEYWELL INC.
Technical Report AFFDL-TR-73-119, Volume I
June 1974
Approved for Public Release; Distribution Unlimited
AIR FORCE FLIGHT DYNAMICS LABORATORY AIR FORCE SYSTEMS COMMAND WRIGHT- PATTERSON AIR FORCE BASE, OHIO
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SECTION II
ANALYSIS APPROACH
This section presents the overview of Honeywell’s work on the definition of the digital flight control system (DFCS) computational requirements for a tactical fighter type of aircraft.
Of specific interest is the sensitivity of aircraft performance criteria to vari- ations in the computational parameters. The approach used was first to generate a comprehensive DFCS stability and performance analysis computer program, and subsequently to apply this analysis tool to a detailed parametric study to obtain computational requirements.
The stability and performance analysis program is briefly presented first.
This program is fully documented in AFFDL-TR-73-119 Volume II. Genera- tion of control laws (synethesis) is summarized next. This is followed by a summary of computational requirements and parametric study.
The computational requirements for the F-4 longitudinal control system are determined by carrying out a detailed parametric study in two levels of system complexity. First, the F-4 longitudinal structural filter is considered. Subse- quently, the overall F-4 longitudinal control system is studied. The results are presented in that order.
Finally, a summary of requirements for digital computation of control laws are given for systems with bandwidths of 6, 12, 20, and 25 Hz. Future growth of control configured vehicles require these broad bandwidths.
STABILITY AND PERFORMANCE ANALYSIS PROGRAM
A computer program (DIGIKON) was generated to facilitate a quantitative analysis of all parameters which affect performance and/or stability. Per- formance includes control and disturbance covariance response, transient and frequency response. Stability includes eigenvalues and gain and phase margins. The parameters include sample rate, word length, computational delays, multisample rates, control filters /laws, aircraft bandwidth, noise, and gusts. The DIGIKON computer program is being used as a tool to develop specifically the DFCS computation rate and control law word length require- ments. This application is shown diagramatically in Figure 1.
MRCHAf I MOOLI
CONTROI 1 ER |
|
KvH |
STRUCTURE |
r +
i
i
►— H
CONTROLLER PARAMl TER S
SAMPLE L
TIME
•v INPUT >» / tARIAUlV A
DIGIKON
sorruARi
. ^ '.OMP ; A ' llj’t«L
> ” wlLAtj
f =
1 l |
J* V. — r* |
|||||
, | |
""If |
0 |
— |
-ir- |
||
POWER CONTENT ANALYSIS |
1 1 1 |
II Ml RESPONSE |
FREQUENCY RESPONSE I'HASl o. GAIN MARGINS |
RMS RESPONSE TO OUSTS |
Figure 1. DIGIKON Software Program for Sample Rate/Word Length Determination
The DIGIKON analysis program (Figure 2) development consisted of two main subtasks. In the first subtask, system modeling software was developed.
This software can handle a general class of digital flight control systems in one of two ways. First, it can construct a set of digital controllers by digi- tizing an existing continuous -controller design for various sets of multi- sample rates. Second, it can accept y-domain i out roller descriptions.
The capability of gemmating a general discrete system model not only develops numbers for a specific tactical fighter system configuration, but also facili- tates in the study of future configurations. This capability also aids in the design of digital control systems, which is outside the scope of this study.
Where specific data is required in this work, the F-4 configured as in the 6H0-I Survivable Flight Control System (Reference 7) is used as the tactical fighter representation. The aircraft model includes a rigid body and three flexure states, actuator and sensor dynamics, and the controller. The discrete controller model variables include sample rate, word length, and computational delay.
4
VEHICLE I'ARAMf TERS
CONTROLLER AND SAMPLE RATE STRUCTURL
CONTROLLER
PARAMETERS
SAMPLE RATES
WORD-LENG1H
COMPUTATIONAL
DELAYS
STATE MODLLEH iSI AMK
'DIGIT AL SYS I EM MODELER'
MULTIRATE ,
HYBRID S > S T E M SIMUL.MON 1
HS-MK
TRANSFOR-
MATIONS
SW/K
MAPPER
OPI'Ml/LR
TRANSTER ruNCHON (SIMK1 )
CONTROL COMPUTER SIMULA! ION COMPK
PEII TORMANCL EVALUATOR
POLES AND ?EROS |
F R 1 OUENCY Rl SPONSL 1 Rl OK |
|
R ANSIEN ' RESPONSE TRESPK |
COVARIANCE *\t. SPONSl i.OVI |
PCRrORMANCE BE SI 1)1 SKA MAP RES! SAMPLE ''ATE
PERFORMANCE FOR SElECIED S'MPIE RATE WORD LENGTH AND CONTROLLER PARAMETERS
Figure 2. Sasic Structure of the Digital Control Software System (D1GIKON)
The development of performance analysis software formed the second part of the Analysis Program task. Subroutines to compute stability and per- formance for the discrete system model were developed. The stability measures are eigenvalues (S, /. and W planes), gain margin, and phase margin. The performance measures are covariance response to wind gusts and random pilot commands, frequency response, power-content analysis, and transient time responses to normal and rapid control inputs and distur- bances. The software uses algorithms based on state-space theory. Each subsystem is characterized by the four matrices (quadruple) (A, B, C, D) for the continuous system and (F,G, H, E) for the digital system. This format facilitates treatment of large-scale system problems. Equations (7), (8), (47), and (48) illustrate the form of how the matrices are used to character- ize a subsystem.
CONTROL LAW GENERATION (SYNTHESIS)
The computational requirements are greatly influenced by the control require- ments. The control requirements basically generate the control laws. Obviously, the two are coupled together (Figure 3).
To determine and validate the computational requirements, one must develop mathematical models for system dynamics and performance analysis. In addition, the control laws should be parameterized with respect to computa- tional parameters (sample-time, word-length and computational delays) to facilitate the DFCS design. In the following, the DFCS design procedures are briefly presented. Conventional control laws are designed by first evaluating the free system performance, and then by deter mining system gains and com- pensators to shape behavior of the system to meet the control requirements. Figure 4 shows this cycle for interactive continuous (analog) controller designs.
The design of digital control laws follows the same pattern (Figure 5). How- ever, more options are available. One starting point is a good continuous control law. This can be transformed into a digital control law by either z -transform or Tustin transform. Another approach is a z-plane root locus design. The free system pole-zeros are mapped and the gains and compensator poles-zeros determined to shape the root locus. The third approach is called z-w plane design (Figure 6). In this approach, the w-transform of the discrete plant model is developed first. Then the w-plane compensators are determined to shape the frequency response to meet control requirements. Finally, the w-piane compensators are transformed back to the z-plane to obtain difference equations for the digital control.
Digitization Versus Direct Digital Design
Following are the advantages of digitization and direct digital design procedures.
FREE
SYSTEM
PERFORMANCE
COMPUTATIONAL
REQUIREMENTS
MISSION |
» |
CONTROL REQUIREMENTS |
— » |
CONTROL LAW STRUCTURE |
Figure 3. Control and Computational Requirements
S-PLANE PUNT (INPUT)
(A B C D '
DUPU
f S-PLANE 1 :0NT NOLLE
FREE SYSTEM PERFORMANCE
INPUT 7 (AcBeCcDc!
(PLANT & CONTROLLER IA8C0) PERFORMANCE
S-PLANE
CONTROLLER
OUTPUT
" We®*’
Figure 4. Interactive Analysis, Design, and Performance Evaluation in the s- Plane
WcV
Figure 5. Interactive Analysis, Design, and Performance Evaluation in the s-z Plane
8
(PLANT
(S-PLANE) DIFFERENTIAL EQUATIONS)
Figure 6. Interactive Analysis, Design, and Performance Evaluation in the z-w Plane
9
Advantages of Digitization
• The requirement starts out with a flyable, continuous controller. Con- tinuous controller provides a strong base for exhibiting effects of the sample time parameter on performance, since it corresponds to the limiting case (i.e. , T 0).
• Controller -digitization algorithms are selected from amcr« those which provide good frequency response and maintains the structural and the stability properties of the controller dynamics invariant with respect
to the sample time parameter. This one-to-one correspondence between continuous controller dynamics (i.e. , lead-lag networks) and the soft ware dynamics (i. e. , corresponding difference equations) provides a good starting point in practical digital controller design for a given sample rate.
• The coefficients of the digitized-controller matrices can be computed efficiently as a function of sample time.
• Sample rate estimates based on this model are on the safe side, and the resulting digital control software is flyable.
• Computational requirements based on digitization can be computed rapidly. In many cases, sharper estimates based on direct digital synthesis methods or digital controller optimizations are not justified for the initial requirement definitions because of the uncertainties in the system parameters.
Advantages of Direct Digital Design --
• W-plane transfer function of the free plant takes into account the delay introduced by the hold unit at the plant input.
• Compensator design with free parameters allows the designer to meet the control specifications with less stringent computer requirements.
An example is given (see page 116) for comparison of direct digital versus digitization synthesis using Tustin algorithm with no prewarping.
COMPUTATIONAL REQUIREMENTS AND PARAMETRIC STUDY
This task includes performing a comprehensive study of digital flight control parameters. Aircraft flight condition, system bandwidth, sample-rate, word length are to be varied, and relative influence on performance is to be examined. The objective here is to define computation rate requirements for a tactical fighter and its sensitivity to DFCS parameters.
1
5
'I
t
10
The F-4 longitudinal control system presented in the fly-by-wire report AFFDL-TR-71 -20, Supplement 2, was selected for the parametric study. First, the F-4 longitudinal structural filter was investigated. Subsequently, the overall F-4 longitudinal control system (open loop and closed loop) was studied. In the following these studies are summarized in that order.
PARAMETRIC STUDY OF "STRUCTURAL FILTER" IN F-4 LONGITUDINAL CONTROL SYSTEM
Parametric analysis by software was carried out to relate the poles and zeros, the frequency response and the rms power response of a structural filter to the computational parameters: sample time and the coefficient word length. The structural filter in the F-5 longitudinal control system was selected for this investigation. For the parametric study of poles and zeros and frequency response, the following parameter set was used:
Sample Time: 0, 1/1000, 1/160, 1/80, 1/40, 1/20 sec.
Coefficient Word Length: 24, 16, 12, 8 bits
For parametric study of rms power response, a first-order prefilter was used with two bandwidths; namely, 200 and 0.2 rad/sec.
PARAMETRIC STUDY OF F-4 LONGITUDINAL CONTROL SYSTEM
The F-4 longitudinal model (aircraft, sensor dynamics, actuator dynamics, and controller) presented in the fly-by-wire report AFFDL-TR-71 -20, Supplement 2, was selected for the parametric study with the DIGIKON soft- ware. The Mach 1. 2, 5000-ft flight condition Rf max) was chosen because of model frequency considerations (highest aeroelastic frequencies). Three bending modes are included in the aircraft model.
Figure 7 shows the four blocks into which the overall model was separated and the interconnections between blocks.
The procedure for data generation for the parametric study is briefly outlined as follows. Starting with the physical equations or the system block diagram, a simulation diagram is drawn. From the simulation diagram, the state equations, summing point equations, and response equations are written. These equations are then programmed for the DIGIKON software. A similar procedure is followed for the controller, sensors, and actuators. After the subsystems have been verified, they are interconnected as shown in Figure 7 by the DIGIKON software.
Parametric analysis by software was carried out to relate the poles and the frequency response of the F-4 longitudinal control system to the sample time of the controller. The following parameter set was used:
11
Figure 7. F-4 Simulation Interconnection
Sample Time: 0, 1/1000, 1/160, 1/HO, 1/40, 1/20 see.
Coefficient Word Length: 24 hits
Following this, a parametric study of F-4 longitudinal control system gust response ratio performance was conducted.
The overall closed-loop F-4 longitudinal control system model was utilized to develop the state and output variance to a gust input. With the continuous gust input, variances are computed with a continuous controller first, and subse- quently digital controllers with T = 1/1000, 1/160, 1/80, 1/40 and 1/20 sample times. The gust was represented by a filtered white noise. A first order filter with bandwidth of 46 rad/ sec was used.
The parametric study was concluded with a brief investigation of computational time delay effects and a new model was developed. In this model, actuator and gust dynamics are modified (a third order actuator and a second order gust filter). The same model is used in the simulation tests.
The following set of computational delays was used for parametric study:
Td = 0, T/4, T/2, T sec.
The sample time was fixed at T = 1/ 40 sec.
The effect of computational delay is studied by computing poles and zeros of the overall closed loop system.
I
SECTION III
MODELING FOR THE DYNAMICS OK MULTIRATE MULTILOOP SYSTEMS
In this section we first briefly present material on the automatic modeling of interconnected dynamical systems for tradeoff studies, kfi lis is followed by the transformations in state space. S, Z and W transforms are con- sidered. The Z-W transformation in state space facilities the interactive design process. S-Z transforms are developed for obtaining the discrete plant representation. Digitization of existing contin^us control laws using the Tustin transformation is also considered. Nexf^we present single-rate modeling of digital control systems by software, •'f'his is followed by the discussion on the multiloop multirate system m^eling. Finally, an example is presented treating a two-rate system with a&nputational delays.
v
4
To perform analytical tradeoff studies of ci/gital flight control systems (or any other control system) one must develop its overall mathematical repre- sentation (i. e. , model). For the linear flight control design, this model takes the form of a set of differential, and /or difference equations.
/
A uniformity in the model form (irrespective of the size or the internal structure of subsystems) facilitates the evaluation of various performance measures in the analytical tradeoff study. One such form is the state variable representation of the overaWroodel.
In the following we brief^ present an algorithm for automatic generation of a model in this form u^ing the physical equations which characterize the elements of the systq/h.
DEVELOPMENT OF THE LINEAR SYSTEM MATRICES FROM THE SIMULATION EQUATIONS
Figure 8 shows a typical longitudinal channel of a tactical aircraft. In general, the simulation equations of this system take the following form: • •
x = f(x, y, x, u) (1)
y = g(x, y, x, u) (2)
r = h(x, y, x, u) (3)
where
x = nx x 1 vector of the output of integrators
14
1
y = x 1 vector of the output of summing points
r = nr x 1 vector of the system variables of interest (response outputs)
u = nu x 1 vector of the external inputs
The functions f, g and h are usually nonlinear. For the linear analysis they can be linearized about a given operating point. In the following, we shall assume that the simulation equations represent the linearized model. In this case. Equations (1), (2) and (3) can be put in the following form:
• •
x = F ’ x + F y+F x + F u (4)
x yJ x u
y = G'x+Gy+Gx+Gu (5)
J x yJ x u
r = II * x + H y+Hx + H u (6)
x yJ x u
and this set of equations can be reduced to the following standard form by algebraic operations
x = Ax + Uu (7)
r = Cx + Du (8)
On the surface, this task appears to be very simple to carry out with paper and pencil. However, for large systems the writing of simulation equations in the format given in Equations (4), (5) and (6) is prone to human error and should be avoided.
In the following, we present an algorithm which automates the transition from the physical equations (analog simulation equations) to the state variable representation given by Equations (7) and (8).
Let us define two vectors as follows:
V = col (x, y, r) (9)
w = col (x, y, x, u) (10)
Obviously, Equations (4), (5) and (6) can be written as
v = F(w) (11)
The matrix coefficients given in Equations (4), (5) and (6) are then obtained by first finding
16
and then properly partitioning it. This term is called the singulation matrix. The sizes of its rows and columns are given respectively by
n = nx + ny + nr (12)
m = 2n+n+n (13)
x y u ' '
The coefficient matrices obtained by partitioning the simulation matrix is indicated in Figure 9.
F- X |
r k |
F X |
F ii |
n t |
C |
G |
G |
G |
it |
X |
V |
X |
it |
, n * |
H. |
H |
M |
M |
I n r |
X |
V |
X |
u |
|
. |
* i |
Figure 9. The Simulation Matrix
The column vectors
w. = 1
l
3F
i=l, 2, .... m are obtained simply by setting
(14)
Wj = 0, j = 1, 2,
. , m, j M
and evaluating (11). This yields the coefficient matrices.
In the sequel, the algebraic reduction process will be described. First, Equations (4) and (5) are written in the following form:
i— i i x. |
-F n y |
[*] |
rF X |
F 1 u |
||
L-°x |
h^7J |
ly |
G L x |
G u J |
x
u
(15)
Then
is obtained in terms of x and u by solving Equation (15).
Then r is obtained in terms of x and u by substituting (15) into (16):
r = (H- | H ) x 1 y'
x
Lyj
+ (H I H ) x 1 u
X
u
(16)
17
These reduction operations are carried out by the computer using the simulation matrix storage space.
The software which implements this algorithm is called STAMK.
Implementation of the Simulation Equations
The analog simulation equations representing the system dynamics [Equations (4), (5) and (6)] are implemented in subroutine SIMK. (The user programs this for his system. )
To demonstrate how SIMK is used, we give the following example.
A simplified short-period equation of an aircraft is given as follows:
0 = (M. J ^ + + + <NUa
= 9 -
«e' 1 e
1
a
U
n
o
nz = <-Za)a+<-Z6e)fie
(17)
(18) (19)
The normal acceleration sensed at station l away from the c. g. is given by:
cl - Crf
n = n + L 0 a z a
(20)
Figure 10 shows the simulation diagram of the short-period equations.
It is assumed that the longitudinal controller configuration is given as shown in Figure 11. It can easily be shown that this controller can be simulated as shown in Figure 12. (For transfer block inputs, SIMKT may also be used as described later. )
In Figure 12,
T
0-rp , a - T , y - a(l~p) l2 2
(21)
18
L_J T*
T?s ♦ 1
r
K |
K |
|
n |
0 |
Figure 11. Longitudinal Controller
I
t
5
!
*
i
:
1
!
i
20
ft. 1 |
T) 1 O |
1 |
(30) |
f) e |
1 |
fi 1 |
(31) |
P1 |
5 e |
(32) |
|
1 2 |
= 0 |
(33) |
|
1 3 |
= n 7. |
(34) |
|
: 4- |
ny = 6, n{, = 3, and nu = 1. |
Subroutine SIMK is essentially Fortran statements of these equations. The right-hand side variables, namely, the integrator inputs (x), the summing point variables (y), the integrator outputs (x), and the external inputs (u), are equivalenecd to the array w(i), i = 1, . . . m array in that order, for example, EQUIVALENCE (TIIKTDOT, W(l», (ALFIX)T, W(2)), etc.
•
Similarly the left-hand side variables, namely, the integrator inputs (x), the summing point variables (y), and the external output variables (r), are equated to the array v(i), i = 1, 2, . . . n in that order. I*’ or example:
• V(l) = MDELE * DEI i MALI-’ * ALE + MT1IDOT * TilETDOT
»• MALI-’D * ALI-'DOT
• V( 2) = TilETDOT - . . .
The parameters such as Vo, Mo , M6e, etc-., are usually equivaleneed to an array of constants, C, which is read in the initialization part of the program for ease in programming.
Generally, a flight control system consists of several interconnected dynamical blocks (i.e. , subsystems). The overall system model is obtained in two steps. First each subsystem model is generated. Subsequently, they are combined using interconnection equations to get the combined model.
Subroutines implementing the subsystem differential equations are named as follows:
SIMKS - SENSOR SIMKV - VEHICLE SIMKA - ACTUATOR SIMKC CONTROLLER
The subroutine implementing the interconnection of sensor, vehicle and actuator (i.e. , plant) is named SIMKP. The subroutine implementing the interconnection of the plant and the controller is named SIMK.
MODELING WITH TRANSFER FUNCTION INPUT
As described previously, the simulation subroutines (i.e., SIMKC, SIMKV, etc.) implement the "differential equations" of subsystem dynamics. To
22
*
k
I
*
$
win.
develop a simulation subroutine for a system characterized by its transfer function, it is required to draw first a state diagram of the transfer function and subsequently to obtain the differential equation from the state diagram. This process can bo automated for rapid and efficient input of transfer func- tion blocks into the DICilKON system.
In the following we present an approach to carry out system modeling by soft- ware with transfer function inputs. The approach consists of two parts: (1)
I '‘or each transfer function block, the corresponding quadruple is obtained, and (2) the subsystems are combined using the interconnection equations and Ihe overall system quadruple is obtained. In the following we discuss each in that order.
Transfer Function and its Quadruple
Consider a system characterized by its output/ input relation:
TO ■ "<s)
v b sn + b -s11 *+...+ b.s * b
Ws) _ _ n n-1 1 o _ i n
n , _ .n-1 , an f U
(35)
as + a , s'
n n-1
+ . . . + a.s i- a 1 o
E
It
There are many ways of realizing this transfer function. (See Appendix B for major realization forms. ) In the following we shall develop the Input I 'robenius form realization and obtain the corresponding quadruple in para- metric form for software implementation.
The long division of Equation (35) yields
b . - |
b n |
a i |
, n-1 , , s + . . . + |
b - |
fb n |
a |
|
b |
n- 1 |
a |
n- 1 |
0 |
a |
0 |
|
= — + - |
L |
n |
nl |
n
n + n-1 .
as a , s + . . . + a, s + a n n-1 1 o
(36)
This can be written as
H = col
C =
D =
I
0 , 0
b - |
b n |
a |
b, " |
b n |
o |
a |
o |
1 |
a |
n f |
n |
b
n
I
a
n -i
b |
b n |
a , |
|
n-1 |
a |
n-1 |
|
n |
I lie transfer function coefficients in Equation (35) form a 2 x (n+1) array as indi- cated below
11s,
J2~l |
1 c _2 |
bl |
b 0 |
||
II |
a n 1 |
a , n-1 |
al |
a o |
(39)
where j is the transfer function block number (Figure 15).
liquations (38) and (39) form an algorithm for obtaining the quadruple of an n-th order transfer function. Subroutine THANSK implements this algorithm.
Figure 14. Input Frobenius Form State Diagram of a Single Input, Single Output Transfer Function
24
■HUHMMiHi
Overall System Quadruple
levelop the overall system quadruple, one must combine the subsystem Iruples obtained as described above using the interconnection relations, iemonstrate the approach taken, consider a block diagram of a system ;aining three transfer function blocks as shown in Figure 15.
' 1
Figure 15. Block Diagram of a System Containing Three Transfer Blocks
Inch block is identified by lour quant i lies: ( 1 ) ;i block number, (1!) IIS army representing the liansl'ei Itmelioii data. Cl) slate u rim number, and H) oil I pul - i n pul pa i r . We note I ha II he i opal a a nd • nil pal a ( I • , uM ) , nl'1, and r< I )) external In llie box are iiirsuhst r ipled va i iabbn, w|,o, can inside l|M bow they are subscripted with i denoting that they ar e internal var iables.
With these definitions, the simulation equations corresponding to the syslem shown in Figure 15 can be written as follows
x(l)- A j x (1 ) ' BjU.O)
x(2) A2xCD ' B./ijCi)) x(3) = A.jXCl) * B3 u.(3)
I )ynam ies
('l II)
25
Internal outputs
(41)
r . ( 1) = CjX (1) + DjU.d)
r.(2) - C9x(2) + D0u.(2) r. (3) = C3x(3) f D3u.(3)
ii.<1) - r.(2) f r. (3)
i i i
u.(2) = u(l) - r . ( 1 )
u.(3) - u(2) r(l) - r.(l)
\ Internal inputs ' (interconnection relations)
^ External output
(42)
(43)
The quadruples ( B., C., D.) i = l, ?, 3 are provided via subroutine TRANSK. The set1 of Equations given above are implemented in a compact form in subroutine SIMKT. The combined system quadruple is obtained via S'l'AMK as described previously.
Here we note that the "form" of the dynamical equations and the internal out- puts are invariant (Equations (40) and (41)). With an additional index indicating the block number, they can be expressed in a compact form for an arbitrary number of blocks. We also note that the variable part described by Equations (42) and (43) have the following structure:
ui = P ri + Qu (44)
r = R ri + Su (45)
The quadruple (P, Q, R, S) appearing in Equations (44) and (45) are called the interconnection quadruple. For this example their values are given as follows:
" 0 |
1 |
1 ■ |
■0 |
0 - |
||
p = |
-1 |
0 |
0 |
. Q = |
1 |
0 |
- 0 |
0 |
0- |
-0 |
1 - |
||
R = |
(1 |
0 |
0), |
s = |
(0 |
0) |
This shows that it is possible to use the same simulation subroutine for model- ing with arbitrary transfer function blocks and interconnections can be used if along with the transfer function data the connection quadruple (P,Q,R, S) is input. The interconnection quadruple of SIMKTC' is not implemented in the DKilKON system.
26
For a demonstration of thi* approach, subroutine S1IY1KTC implements liqua- tions (40) through (43) for the F-4 continuous controller. This is presented in Appendix A.
TRANS FOR MATiONS IN STATE SPACE
Discrete models for both the control and the plant are required to perform sample rate and wordlength tradeoff studies. We present two methods for obtaining these models from their continuous representations. All discrete models are expressed by the following standard- form set of difference equations
x(k+ 1 ) F x(k) * (I u(k) (47)
y(k) II x(k) i K u(k)
where
x(tj_) = x(k) v ( t k ) - y(k) u(tk) " u(k)
The /-transform is used to develop the discrete model of the plant. The Tustin transform (T- transform) is used to develop the discrete model of the control- ler from its continuous model (digitization). The /-transform can also be used on the controller.
To facilitate direct digital design, the w-transform is also developed. The results are summarized briefly in the following paragraphs.
DISCRETE MATRIX MODEL FOR THE PHYSICAL PLANT
Referring to Figure 16, we see that there are two kinds of inputs to the plant.
1) Continuous inputs (wind gusts and other analog disturbances, r| )
2) Piecewise constant inputs (from the zero-order hold units, x^)
The problem here is to find the exact response of the plant states at sample points as well as all other intersample time points with these inputs.
The analysis starts with the physical plant continuous matrix quadruples (Ap, Rp, Cp, Dp). This quadruple is obtained by software (STAMK) from the simulation equations of the plant as discussed above. The physical plant equations are given by
27
TO DIGITAL COMPUTER
Figure 16. State Diagram of Physical Plant Including Hold Elements
x - A x + B u (49)
P P P P P ' '
y r C x M) u (50)
P P P P P
The state response is given by
A (t-t. ) t A (t-s)
x (t) = e * x (k) + J e p R u (s) ds (51)
P P t P P
lk
where x (t. ) = x (k).
P k p
In Hie following, the discrete matrix model dor the physical plant with pieee- wise constant inputs is developed. [For the response to both kinds of inputs, see liquation (T17) or page 109. ] For this ease 1 he state response of the plant is given by:
A (t-t. ) As \
Xp(t) = e p Xp(k) + J' e p Rplds j up(k) (52)
where o 5 (t-t^) s T.
At sample points we have
Vk+U = Vp(k’ * GpiuPi<k>
where xp(tk+1) = xp(k+l)
28
t- „ —
i rrtni-ir i bti t i iifc'Miiiia Wmiti tftittflTi'
Therefore, by tins procedure we eliminated the integral given by Equation (55) or the relationship given by (56) which is unduly restrictive.
In summary, we first form Equation (60) then evaluate (61) as described below, and, finally, partition Kj as shown in (62). This yields the sought matrices F and Gpj.
AT
To compute F = e , we use the following algorithm:
AT _ ( . -AT]"1 (. . AT)
; = \ 1 + e 1 1 + e
m (AT]_ + El"1 (I+I+AT+*4^- + . . + Eo]
= [1+1- AT + (jp- + . . ^-l)m ~r— + E0]_1 (1+I+AT+—7
where m is the maximum power used in the rational approximation. For m = 3 this yields
eA1 = F + 0(T5) P
where
v 1 r1 '•+
. - AT (AT)2 (AT)2
12 4 12
AT (AT)2 + (AT)3
*2'1 + T*"T“ ~TT
The terms appearing in Equation (64) are recursively computed. An option is available so that the power series expansion
AT ^
F(T) = I + AT +(^y-) + ... (69)
AT
of e can be computed for specified numbers of terms as well. The algorithm specified above is implemented in Subroutine EXPK3.
Selection of Transition Time
The transition time T used in EXPK3 is computed from
T = 2'k T k s
30
where
Tg = Sample interval over which matrix expotential is computed k = integer 1
The subinterval index k is predicted using the maximum eigenvalue of the continuous system matrix A. The actual value of the parameter k and the intersample time interval, T|t, aic subsequently obtained using a relative error criteria.
S nice
l'(T ) = (ITl . )]2 s k
tin’ successive values
(7 1)
.k+1
and lK(T.)r are computed, and the relative error
on each element is found. The index k is incremented until the maximum relative error becomes less than a specified number. Non-normal exit with a proper message occurs if k exceeds its limit, or if the relative error can- not be reduced further. This computation is followed by the eigenvalue and steady-state gain checks. The steady-state gain is defined as the steady- slate value of the state vector of the system, subjected to unit step input, if it exists (i.e. , x 0 for continuous systems; *k for discrete systems]
Since the sampled states and continuous states must have the steady-state value we get tin* following gain check equation
■ A~ 1 H (1 - I’)”1 G
(72)
The subroutine i:\IM\2 implements the above algorithm.
The eigenvalues are computed both in the /-plane and the s-plane. The eigenvalues s^ of the A matrix is transformed to /-plane using
:n
•-
'
s.T - k s
and subsequently compared with the eigenvalues of FfT l ai.. +v. .
values zk of FIT,) are transformed tofteTmage-s-pSe vta ’ *“ eig*n-
sk = Tf-( l°i I \ I + j ek) where \ * | *k | ej0k
and compared with the eigenvalues «k of A. As is well known, this inverse process is not one-to-one unless the half sampling frequency
s = n
2 T
the8 eigenvalue of*A ,m fc'mam c^ses tota'cond'it-*1* ^arge^t ““aginary part) of program computes the foldover index q from tee^ reUttai <foldover)- The
** A
U) = ID + q iij
* o /nr 1
where
id = corrected frequency
id = computed frequency from Equation (73)
iDg = sampling frequency from Equation (74)
Both corrected as well as folded frequencies are printed out for comparison. This finishes the description of the algorithm for computing the pair (F, G).
sabme!USThatTs.lhiS °aS“ ‘he °utput equation (se« Equation (48)) remains the
HP CP (76)
E = D
P P (77)
Then at sample points, the state of the plant is described by
vk+i) = FPvk) + GPiVk)' y°> * xp0 <78)
yp(k) . 1 ycp(k) + Epup(k) m
32
This finishes the description of the algorithm for obtaining the discrete matrix quadruples (Fp, GpJ, Hp, Ep) corresponding to the physical plant or
control plant driven by piecewise constant inputs.
Again we note that the discrete matrix quadruple (Fp, Gpl, Hp, Ep) of the
plant as generated above is a function of the sample time, T . The sub-
0
routine which implements this algorithm is called subroutine EXPK. It is fully documented in Section IV of Volume II.
33
(80)
Taking the z-transform of Equations (78) and (79) gives
(zI-F ) X (z) = z x (0) + G U (z)
P P P P P
Yp<z) * HpXpU) + Vp(z> (81>
Assuming zero initial conditions [i. e. , Xp(0) = 0] we obtain the input-output relation of the system in the z-domain as follows:
Yp(z) = [IlpUI-Fp)'1 Gp + Ep] up(«> (82)
The z-transfer function between the i-th output and the j-th input is then given by:
Y^z)
tTTzT =
J
H..(z)
P1J
(z!-Fp
(83)
where h . and g . are the i-th row and j-th column of the H and G matrices, pi fapj P P
respectively.
We note here that the presentation of design methods and procedures is outside the scope of this work. However, we point out that the available software in this program can be used to facilitate the design.
For "direct digital design" in the z-domain, for example. Equation (83) (or (82)) becomes the starting point of the design (i. e. , the z-transfer function of the free-plant). The poles and zeros of these expressions are found by software (POZIO as will be described later. Subsequently, compensators are designed using the ropt-locus in the z-plane.
DISCRETE MATRIX MODEL FOR THE DIGITAL CONTROLLER
To develop a discrete time model for the continuous controller dynamics, the
matrix version of the Tustin algorithm is used. The z-transform could also
be used (as above for the physical plant) to obtain somewhat different results.
The analysis starts with the continuous controller matrix quadruple (A , B ,
c c
C , Dc>. (This quadruple is obtained by software (STAMK) from the simu- lation equations of the controller as discussed in Appendix A. )
The controller equations are
x = A x + B u c c c c c
y = C x + D u ’'c c c c c
Transforming Equation (84) gives Xc(s) Msl-A^"1 BcUc(s)
(84)
(85)
(86)
34
This can be written as
Xc(s)
=T A T
ST T C
T1' —
* |
B T \ |
-C- |
|
2 J |
u (s) c
sT z~l
Now replacing -g- by (Tustin's Rule), we obtain
Xc(z) =
i A T Z“1 T c
z+1 2
-1
B T
~§” uc(z)
(87)
(88)
Clearing the fractions and rearranging.
Xc(z) - (zl - Fj-1 F2)-1 (zl+l) Fj^GjU^z)
where
A T
I •
(89)
(90)
F = r 2
I +
A T c
B T
G1 = ~ 2 —
(91)
(92)
We note here that F. and F are analytic functions of A . Therefore, they
commute with A , 1 £
c
From Equations (90) and (91) it follows that
I=2F1'1-F1-1F2
(93)
Substituting this into the second term of (zl+I), Equation (89) becomes
X
:“>■ [(F1’1Gj ) + (*I - F,'1 Fj)*1
(*)
(94)
35
M Ifl
rr-
Substituting this into Equation (85) yields
YC(Z) =[cc(zl - Pr1F2)-l(2F1-2G1|+|ccF1-1G1 +Dc|]uc(z) (95)
The transformed system has a new set of states which we the subscript d. |
shall identify with |
Letting |
|
Fc=F.'lF2 |
(96) |
Gc - iFj^Gj |
(97) |
Hc=Cc |
(98) |
Ec * Dc + CcFl"loi |
(99) |
one can write the state equations of the digital controller as follows:
(100)
yc(k) =Hcxd(k) + EcUc(k)
(101)
We note that Equations (100) and (101), with matrices defined &y Equations ) through (101), have the transfer function given in Equation f 95).
The state diagram of the digitized controller is shown in Figure 17.
Figure 17. State Diagram of the Digitized Controller
36
' -
In Figure 17,
x^Uc) ■ State of the digitized controller
xc(k) = "Digitized state" of the continuous controller
yc(k) = "Digitized output" of the continuous controller
We also note that the controller matrix quadruple defined as (F , G , H , E )
0 0 0 0
is a function of the sample time, T (see Equations (90), (91) and (92)).
The system quadruples defined by Equations (96) through (99) are implemented in subroutine SWZK. It is fully documented in Section IV of Volume II.
The Tustin transfer function is given by Equation (95). As with the physical plant model, the poles and zeros of this function can be found using a sub* routine (FOZK) with the developed quadruple.
Steady State Gain
The steady state response of x (k) to a unit step input is obtained from Equation (94) as follows:
* [vs + (ji - vsr 2Fr2°i]
Z = 1
This can be written as
*cas= [Frl+2Fr1(Fi-F2>'l]Gi
Using Equations (90),(91), and (92) with (103) yields
{v1
I + -
AT -l] B T
(102)
(103)
(104)
A Tl-1
Factoring -I and making use of Equation (90) finally yields
x = -A “*B (105)
css c c
37
This shows that the steady state gain under the Tustin transformation is invariant. If the continuous system is prewarped for locating the critical frequencies, a correction to the gain term is made to maintain the steady- state gain invariance.
Prewarping for Pole Placement
Consider the following conformal transformation
(l+Sf
(106)
We can define a matrix function of a matrix A, corresponding to Equation (106) as follows
F(A) = I-^”1 1 + ^
(107)
Let the eigenvalues of A be {s^j k = 1, ... n. Then the eigenvalues of F(A) are given by
3 jT'-i 3kT
5k(T) = 1 • HH 1+-t-
k = 1, . . . , n
(108)
This relation shows that when A is a stability matrix (i. e. , all eigenvalues are in the L. H. Plane), then eigenvalues of F are in the unit circle. We note that the same is true for the matrix
F(A, T) = e
generated via the transformation
f(s) = e
(109)
when the eigenvalues s T
zk(T) = e c
(110)
38
P
I
i
\
For each fixed s^, the locus of Equations (108) and (110) as a function of sample time parameter T shows that
iim ?k(T) = -1
(111)
T -* ®
and
iim zk(T) = 0
(112)
T —
Also, 5k(T) does not cross the real axis in the range
0 < T < • (113)
This implies that the poles of ?k(T) always remain under the half sampling
frequency (tt/T). Therefore, under this transformation, the "system modes" do not foldover for any sample time. The penalty we pay for this nice property is the shift in frequency. The shift can be compensated for a given sample time T. This is called prewarping of a continuous system. We note that when the system is prewarped to maintain critical frequencies, the non- folding property of Tustin is lost.
Let A be the prewarped transition matrix corresponding to a continuous controller matrix A. Let F be the corresponding discrete system transition matrix
defined by Equations (90), (91) and (96). If F is to have the same poles it
AT
must be similar to e . The simplest case is:
F =eAT
(114)
or
I -
AT -1
I +
AT
= e
AT
(115)
39
Hi ~ri n i i
Mil*
miit
Solving this for A yields
- 2 AT
A = y tanh 2
On the other har d, from Equation (93) we obtain
•1 I + F
Substituting this into Equation (89) and introducing a gain matrix K
H(z) = K(zl - F)"1 (--£) (z + 1) BT/2 The steady state gain invariance requires
H = K(I - F)"1 (I + F) BT/2 = -A_1B css
Solving for K yields
~ ~ -1 ax -1
K ■ (I - F) (1+ F) (" ^)
Substituting Equations (120) into (118) yields
H(z) = (zl - Ff 1 (F - I) A"1 B 2-— using Equation (114) gives
H(z) = (zI-eAT)-1(eAT-l)A-1B4i G = leAT - iIa"1 B
Then Equation (1J2) can be written as
40
Now by inspection we can write discrete quadruples corresponding to the prewarped Tustin transformation as follows :
F=eAT |
(125) |
G = (eAT - I) A-1B |
(126) |
H «= (C + CF)/2 |
(127) |
E = D + ~ |
(128) |
Note close resemblance between above and the plant discretization given by Equations (52), (54), (76), and (77).
State Model of the Discrete System in the w-Plane
Direct digital ccntrol synthesis in the z-w plane calls for algorithms for finding the w-plane transfer function from the z-plane transfer function and vice versa. In the following we present one such algorithm based on a systems approach. The development starts with the discrete system matrix quadruple (F, G, H, E). (This quadruple is obtained by a software (STAMK) from the simulation equations of the discrete system structure as discussed above.)
The system equations are: x( k+1) = Fx(k) + Gu(k) y(k) = Hx(k) + Eu(k)
Transforming Equation (129) with zero initial conditions yields: zX(z) - FX(z) + G U(z)
The transformation to the w-plane is defined by
, - 1 + w 1 - w
(129)
(130)
(131)
(132)
The inverse transformation is then given by
(133)
Substituting Equation (132) into (131) yields:
X(w> = - wl - j-F^1 F2jJ 1 Fj'1 (wl - I) G U
(w)
(134)
where
Fj = (I + F)
F2 = (I - F)
(135)
(136)
From Equations (135) and (136) it follows that
-1,
I ■ ~Fj-*f2 + aFj'1
(137)
Substituting this into the second term of (wl -I), Equation (134) becomes
"1
Xlwt-tlwI-l-F^F.,)] aFj^G-F^GjUl*) (ls8)
w-plane-1"8 *h‘* “t0 E’uation (130» yield. the input-output relation in the
-1
Y(w) = (H [»I-(-Fi-‘f2)J JFj^G-HFj^G+BJWw) (139)
tte%urb“cr%™wd SyStem haS “ "ew set ot sUtea wtuch we shall identify with We define w-plane quadruple as follows:
A = -F _1
w
1 F2
Bw = 2Fl G
Cw = H
Dw = "H ¥i~1g +e
(140)
(141)
(142)
(143)
42
LhefoTiows:qUa“°n3 °f fhe dUCrete SyStem “ the W-P>“« now be wr .ten
dx
w
~j7 — = A x + B u dt w w w w
yw " Cwxw + Dwuw
(144)
(145)
a40)Oiero^^?^ra (1*4) and (145) with matrif'es defined by Equations (140) through (143) have the transfer function given in Equation 139
The state diagram of the discrete system in the w-piane is shown in Figure
18
Figure 18. State Diagram of the Discrete System in the w -Plane In Figure 18,
xw = state of the w-plane system x = state of the discrete system in the w-plane y = output of the discrete system in the w-plane
quadruple Ta^b™8 C th^tr.ans^ function are obtained via POZK using the (140) throughTl 4 3T arp '' 7 " qUadrUpie defined by E<^tions
back to the -plane is cSeToTin • The transf°™*tion
Thedsummary of the results on trVns^
43
To demonstrate the application of these equations we present a simple example in the following.
EXAMPLE
Consider the following z-plane transfer function
xt /-* _ 3(. 368 z + .264) Ho(z) ■ (z-n (z - : 56-sr
(146)
This can be put in the following form
H (z)= 1. 104 V--J17-4J
° z - 1. 368z + .368
(147)
Figure 19 implements this transfer function as an Input- Frobenius form.
Figure 19. State Diagram of llQ(z)
49
The discrete system quadruple (F, G, H, E) is obtained from Figure 19 by inspection, and given as follows:
0 |
1 |
0 1 |
||
F = |
11 o |
|||
-.368 |
1.368 |
“1 -<r o • 1 |
H = (.7174 1.), E = 0 (148)
It can easily be shown that the transfer function
Hn(z) = H(zl - Ff 1 G + E (149)
evaluated with Equation (148) is the same as Equation (147).
Now transforming quadmple data in Equation (148) to the w-plane using the appropriate equations given in Table 1 yields the w-plane quadruple data:
w
"o |
1 |
"o" |
||
r |
, B = |
|||
0 |
-.462 |
' w |
1 |
, C = (.69299 .976), D
w
It can easily be shown that the transfer function
H. . (w) = C (wl - A f 1 L + D 11 w WWW
evaluated with Equation (150) yields
2
Hn(w) = -1. 14168
w - .3 9 3 w - . 607 w2 + . 462w
This can be written as
1.5(1 - w) (1 +
w
Hj j( w) =
7m
The same result is obtained by substituting 1 + w
z =
1 - w
w
1.14168(150)
(151)
052)
(153)
(154)
into Equation (146) and clearing the tractions. For large systems, the substitution approach is not suitable lor automatic evaluation d Equation (153) due to the associated algebra. The quadruple transfor- mation approach on the other hand is simple, accurate and suitable for large scale systems. We note here that Equation (154) and its inverse.
4t>
(155)
given by Equation (155), are one-to-one transformations. Therefore, if the w-plane quadruple (Aw< B^, Cw> D^) given in Equation ( 150) were trans- formed back to the z-plane, the result would be identical to that of Equation (148).
Figures 20 and 21 demonstrate this fact, using the F-4 digital controller for
the sample time T = 1/40 sec. In figure 20, the controller quadruple
( F, G, H, E) in the z-plane is entered, and the w-plane transform (A , B , C , D
is computed. Subsequently, this data is entered and its z-plane w w v/ w
transform is computed as shown in Figure 21. As expected, the output
data in Figure 21 is equal to the input data in Figure 20.
OVERALL SYSTEM MODELING FOR SINGLE RATE SYSTEMS
Having the discrete model for the plant and for the controller, we would now like to develop analytically the discrete model of the plant-controller system. We need this for trade studies of sample rate and word length. The complexity of this model depends upon the form of the control (algebraic or dynamic) and the number of different sample rates in the combined system. (We neglect computational delay effects here for simplicity. These are considered below. ) We show how to construct the overall discrete system model for a single sample rate here, and consider the extension to multiple sample rates in the next section.
Algebraic Controller
Figure 22 shows the general block diagram of the single-rate system under consideration.
The plant has the usual discrete representation
xp(k+n = Fpxp(k)+Gpup(k) (156)
Vk) =HpVk) + EpVk) (157)
The controller, for the algebraic control system, has the form
rc(k)=Kuc(k) ( 158)
47
I
I
• o CO •
♦ • • • I
K ■» t* P» <Vj
5|S SS2
w* ® C"*» #
•|*
•*1 — «|«» M o • !® o<# *!o
esi
f** *v. © ® • o 2 - ^
n n r\j ^ mc
— o % o #
• o o o o
I
I
I ^
• #'0 c«
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I # K i * iT ' ^
48
OUTPUT DAT*
Transformation
"ODE = T?9 SAMPLE TIME = .?*999F-91 SEC.
INPUT DATA: W-PLANE MATO I X OUAOPUPlE <A*.84.C4.04)
OUT°UT DATA; 7-PLANE MAtPIX OUAOOUOLE <F.f,.M.EI
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51
dBBauii^
Figure 21. Transformation of w-Plane to z -Plane (Page 2 of 2)
h
<k)
mu
Figure 22. Block Diagram of a Single Sample Rate System That is, the control system box in Figure 22 contains the gain matrix K. We
also have from Figure 22 that
Up( k) = r^(k) (159)
uc( k) = u( k) - r (k) (160)
Our objective is to reduce Equations (156) through (160) to the form
x(k+l) = F x(k ) + G u(k) (161)
r(k) = H x(k) + E u(k) ( 162)
where u(k) is the sampled version of the input u(t). This is the overall discrete representation of the system of Figure 22. One easily solves Equations (156) through (160) to obtain
x(k) |
■ V1" |
(163) |
r(k) |
= rp(k) |
(164) |
F |
= F - G KM H P P P |
(165) |
G |
= G K M P |
(166) |
H |
= M H P |
(16?) |
E |
= M E K P |
(168) |
52
■frMrtiMYh
if:
with
M = [I + EpK] _1 (169)
In words, the state and response of the single sample rate system with an algebraic controller are the state and response of the plant. The matrix quadruple (F, G, H, E) is computed from Equations (165) through (169).
Dynamic Controller
In this case the plant in Figure 22 is represented by Equations (156) and (157), and the control system is given by similar expressions:
x (k+1) * F x (k) +G„ U (k) c c c c c |
(170) |
rc(k) = Hcxc(k) ♦ Ec uc(k) |
(171) |
The relationships of Equations (159) and (160) still hold. Our objective is to derive the overall discrete representation [Equations (161) and (162)] for the system described by Equations (156) and (157), (159) and (160), aid (170) and (171).
The discrete overall representation is much harder to obtain for the dynamic controller than for the algebraic controller. One finds that
x(k) = col [Xp(k), xQ(k)] (172)
r(k) F =
G = H =
* col [rp(k), rc(k)]
1 |
1 CM rH Ph |
LF21 |
F22 J |
r°ii |
|
i CM <g j |
|
'Hll |
«12' |
-Hll |
H22- |
(173)
(174)
(175)
(176)
(177)
Ft 7 = F - G M E H 11 P p c p
F12 ■ G1 M Hc
F21 = ^c'1 ‘ Ep M E JH
c- p
F22 ■ F * GEM H cep c
G. * G ME 1 p c
C2 Gc H - E„ M EJ
p c‘
H11 * U * Ep M Ec] Hp
H12 ■ EpMHc H21 * * M Ec »p H22 = M JIc
E. - E M E
1 p c
E9 = M E
2 C
(178)
(179)
(180)
(181)
(182)
(183)
(184)
(185)
(186)
(187)
(188)
(189)
M ■ [I + E E 1 c pJ
(190)
“s z ztj2 c cf°tr
HatE?fi9blh £ Z
co„t ro^d GAt Vh o°nf «bh1hdLhr.al SST. tZl 'Hi™* analy“C
54
Parametric Interconnection Model and Interconnection Quadruple
The previous example leads us to the parametric interconnection model with interconnection quadruple (P, Q, R, S). This is illustrated in Figure 23.
(AUGMENTED DYNAMICS)
(F|,Gj/Hj/E|)
'
Figure 23. Parametric Interconnection Model
Xj = col(Xp, x£) = augmented state r. = col(rp, rc) = augmented internal output
Uj = col(Up, uc) = augmented internal input
u = external input
r = external output
Let (F-, G^, H., E^) be the augmented quadruple given as follows:
r F 0 ”1 rG 0 -| fH 0 1 TE 0
P P P p
Fi - . Gj - . H = . E .
L° FcJ L° GcJ L° HJ p L° Ec
(191)
The following system of equations describes the overall system model:
x+ = F. x + G.u. i l k |
(192) |
|
r. = H. x + E.u. l l i k |
(193) |
|
ui = P r( + Q u |
(194) |
|
r = R r. + S u |
(195) |
|
Solving Equations (193) and (194) in terms of x and |
u yields: |
|
r. = (Iri " E. P)_1(H. x + E. Q u) |
(5) |
(196) |
u. = (Iui - PE.)"1 [PH.x+Qu] |
(6) |
(197) |
Substituting Equations (197) and (196) into (192) and (195) yields the overall system quadruple in the form of Equations (161) and (162) |
||
where |
||
F = [Fl+Gi(Iui-pEi)'lpHi) |
(198) |
|
G - [G1 (Iui - P E(f 1 Q] |
(199) |
|
H = [R <Ir. - E(Pf 1 Hj] |
(200) |
|
E = [R<IH - EjP)"1 EjQ +S] |
(201) |
OVERALL SYSTEM MODELING FOR MULTI- RATE SYSTEMS
In the following, an overall state model is developed for an algebraic digital control system (Figure 24) having two different sample rates (inner loop and outer loop rates).
The following equations are derived based on Figure 24 with the digital con- troller (or the digital Computer which implements the control law) which operates on an input sequence of sampled information to produce an output sequence of sampled output^ This sampled output is converted to the piece- wise constant signal by the holds HI and H2.
i
56
i
h? ")
"l “i
li = An^fljUj
'2mCi
• £OMPUTLr
'p — <T^o— li(
»«•> i -
O -H K,
i
Figure 24. Two-Rate Algebraic Control System
It is assumed that the outer loop sample time T2 is an integer multiple of the inner loop sample time Tj. In this case T2 becomes the program period, and
the transition equation at the sample points kT2,k = 0, 1 becomes
stationary. This equation is given by
x(kT2 + Tn) = Fx(kT2) + G fr (kT2)
(202)
F ■ <F10 + G2K2H2>- 0 = mf- 2 1
(203)
G2 = (Fj + . . . + Fj + I)G,
(V
F1 = (F1 +giK1h1)
(204)
(205)
Fl-e
(206)
•T1 _As„ AT
J1 As 1 - 1
n e Bjds = (e -I) A Bj (if A has no zero eigenvalues) (207)
57
(208)
As_ , e Bgds
- I) A^Bj
The stability properties of the system are obtained using F.
DELAY SYSTEM MODELING
A block diagram of a model of a system with an algebraic controller and a computational^ de^av is shown in Figure 25.
HOLD .
Figure 25. Computational Delay Model
It is assumed that the control input to the plant is updated T state is sampled as shown in Figure 26. c
seconds after the
■i
i
1
58
Table 4 shows the sequence of transitions and corresponding transition equation for each transition.
Table 4. Sequence of Transitions
Time |
Description of Event |
Transition Equation |
*k |
Beginning of ■ new program cycle |
[yv- xh(tk>) |
*k+ |
SW1 samples *p (t^> and computation of controller output starts |
rc(tk> ’ K W |
The plant state xp undergoes a continuous transition |
ytc,*vTcVvtoP(Tc)xh(tk)*t,tc |
|
*c* |
SW2 transmits the computed output to hold unit xh undergoes a discrete transition |
VW * rc(tk‘ |
‘c+^'k+l |
xp undergoes a discrete transition |
V‘k+l)s VT*Tc,XP(tc)+Gp(T-T.)xh(tc+) |
Let us define x * col (Xp, x^)
Then using Table 4 we can write
(209)
(210)
(211)
Substituting Equation (210) into (211) yields
x(,k+i) =
F(T-T ) c |
G(T-T )“1 c |
>(tc) |
°<TC>1 |
|
O |
K |
0 |
x(tk) (212) |
or
X(W
f F(T) + G(T-T )K
F(T-T ) G(T )
L K
Noting that
G(T-T ) = G(T) + F(T) G(-T )
c
F(T-Tc)G(Tc) = -F(T) G(-Tc) Equation (213) becomes
X<V
x(kT+T) = F(T, T ) x(kT)
v
where
F(T, T ) = |
Fp (T) + AH(T,Tc)K |
-AH(T,Tc) |
c |
K |
O |
and
Fp(T) = eAT
Gp(T)'^e SBds = (eAT-l) A-1B (if A has P "P
(213)
(214)
(215)
(216)
(217)
P o F_(t) = F (t) + G (T) K
(218)
no zero eigenvalue) (219)
(220)
AH(T,Tc) = Fp(T)Gp(-Tc)
(221)
60
I
1
Y,*ilT**W^'Z‘-iatv *v«v
It la interesting to note from Equation (209) that the order of the dynamics which describe the state is increased from n to n+m where n is the order of the plant and m is the number of control inputB to the plant. As the compu- tational delay Tc is reduced, the plant state Xp(kT) becomes less dependent
on the hold state xh(kT). In the limiting case, this dependence becomes zero. Tne perturbation term given in Equation (221) is easy to compute and
l im AH(T,T ) = 0
V°
(222)
For small T , A H becomes proportional to T and is given by: c c
AH(T,T ) = -F(T) B T
v> v
(223)
The effect of computational delay on the stability of the digital system is studied by using Equation (217).
GUST RESPONSE MODELING FOR SAMPLE TIME EFFECTS
This model is used to determine the gust response (i. e. , normal acceleration cross-range error, etc. ) as a function of sample time. The system specifica- tions (i. e. , ride quality, landing specs) impose limits as to how large the sample time can be without exceeding these specifications. In the following the gust response is determined not only at the discrete sample time points, but also at all other time points (intersample covariance) as well. The inter- sample covariance is periodic, with periods equal to the program period. The n-th order model is shown in Figure 27. —
COMPUTER
l&t
COMPUTER
A, BlUp
T I I
Figure 27. Gust Response Model for Sample Time Effects
j
k;
The physical plant (aircraft) is described by
x = Ax + BjUp + B2w
(224)
where
w = white noise gust input vector
up ' KVrk> tk*t<*krt
(225)
We assume that the open loop transition matrix A has no zero eigenvalues. (This is not a necessary condition. It simplifies the analysis. ) Then the intersample covariance response is given by
X(t) = F(t) X(t. ) F '(t) + J eA(t_s)B_WB/2 eA (t‘8)ds
(226)
where the prime indicates the transpose, and where F (t) = CeAt + (eAt - I)A“1BK]
W = E{ w w'}
\ * 4 < Vl
(227)
(228)
The noise inputs, w, are assumed to be stationary. In this case W is a constant matrix. At the sample points (t = t^) Equation (226) becomes
stationary; and if F(T) is a stable matrix, then Equation (226) has a steady state solution given by
X = F(T) X F'(T) + V(T)
(229)
where
V(T) =
1
/
eAsB2W b' eA sds
(230)
This solution is computed by using the following iterative equation and fast partial sum technique.
X*1+1) = F(T)X(l! F'(T) + V(T), X(o) = V(T)
(23!)
62
Once X is found, the intersample covariance response is obtained from Equation (226) as a function of t for tfc * t < tfe+1. The following example is
presented to demonstrate the application of these equations.
Figure 28. Simple Digital Control System with Continuous Disturbance Input
In this example.
-T
-T
F » e .* , G = (1 - e 1 ), F = F - GK = (2E-T - 1)
and
V(T)
■ / e‘s °«
g2 e“s ds = -4j- (1 - e ”2T)
-2T,
(232)
(233)
The steady state variance at sample points is given by X = (2e”T - 1)2X + 1L"?2 2T). „ 2
or
8 s
(1 - e~2T) 2
^rrx a-
2[l-(2e” 1 -1) j S which reduces down to
(234)
(235)
Y _ 1 + e1 2 Xss 5 °g
(236)
This shows that as sample time goes to zero (toward continuous closed loop control) the output variance becomes
2
X = (237)
We also note that the open loop output variance is
Xopen ■ \ <238>
Per unit steady state output variance at sample points takes on the following values as a function of sample time [Equation (236)]:
T = 0 0.5 1 1.5 sec
XDQ = 0. 25 0. 33 0.46 0.69
D S
The intersample response can be computed from Equation (226) for each fixed sample time
Xxt) = F(t) Xas F(t) + V(t) (239)
-2T
X(t) = (2e~t-l)2 X8a + ■ ) ag2 OstsT (240)
This response is periodic with period T [see Equation (229)].
X(T) = Xss
The periodic extension of Equation (239) constitutes the meansquare response of the system for all times. This response is plotted in Figure 29.
DISCRETE SYSTEM MODELING BY SOFTWARE - DISCRETE SINGLE- RATE SYSTEM
In the previous paragraphs we developed models by analytical means. We now present a procedure for obtaining an overall discrete system model by soft- ware. First we develop a single rate model with no delay. Following this, we describe a multirate model with computational delays. Figure 30 shows the block diagram of a single rate system.
64
Figure SO. lilock Diagram of a Single Rate System
We define the two vectors
V = col |
x (k+1), x (k+1), r (k), r (k), u (k), p c pep |
uc(k), r(k)J (241) |
w = col |
x (k+1), x (k+1), r (k), r (k), u (k), p c pep |
uc(k), Xp(k), |
xc(k), u(k)J |
(242) |
|
Then equations describing the system are written in SIMM. |
the form (Subroutine |
|
X (k 4-1 ) P |
= F x + G u P P P P |
|
x (k+1) c |
= F x + G u c c c c |
|
r (k) P |
= II x + E u P P P P |
|
r (k) c |
= H x + E u ' c c c |
(243) |
Up<k> |
= rc(k) |
|
u (k) c |
= u(k) - r (k) |
|
rj(k) |
■ rp(k) |
|
r2(k) |
■ rc<k> |
|
Subroutine STAMK is used as described previously to find the overall system quadruple (F,G,H, E). |
In the above development, we tacitly assume that the plant and controller inpuls are updated at the same time. In this case, the transition points occur at the beginning and at the end of the program period. (No discrete transition exists in the
66
l
interval. ) This is the simplest structure (single-rate) in computer-controlled systems. In practice, one often encounters two-rate and three-rate systems with computational delays. In these systems multiple transitions take place within the program period. In the following we present a procedure for modeling such systems by software.
DISCRETE SYSTEM MODELING BY SOFTWARE - MULTIVARIABLE MULTIRATE SYSTEM MODELING WITH COMPUTATIONAL DELAYS
In general, the digital control systems are constructed by interconnecting four types of dynamical subsystems: (1) continuous dynamical subsystem (Plant); (2) continuous holding subsystem (D/A output); (3) discrete- dynamical subsystem (control law software); and (4) memory holding subsystem (describing the delayed variables due to computations within the digital con- troller).
Behaviour of the state transitions corresponding to these subsystems are shown in Figure 31 with a typical feedback system interconnection.
r'
6
DISCRETE
CONTROLLER |
um |
MEMORY HOLD |
un |
P/A HOLD |
UP. |
PLANT |
(kc) |
fe |
V |
rm |
(x„> |
rn |
(y |
Figure 31. Time Behavior of State Transitions
67
To develop a ni:.thomnti(*Rl model for such systems which is valid for all limes, a hybrid-state is introduced representing the overall system by
x = col (Xp. xc. xh, *m)
where
xp ■ Physical plant state (output of integrators)
x = State of the zero-order hold units describing the piecewise h constant inputs to the plant
xc = State of the digital controller
x = State of the memory units (describing the delayed variables
m due to computations within the digital controller)
Table 5 shows the form of the transition equations and corresponding quad- ruples.
Table 5. Forms of Transitions of Dynamical Subsystems
Input-Output
Representation
Equations
Quadruples
Remarks
Physical
Plant
x = F x +G u P P P P P
r * H x +E u P P P P P
(F , G , H , E ) P P P P
Represents interval transitions
Hold Units
x h ' Uh
(O. I, I. O)
Represents discrete trnns it ion
Controller
x = F x +G u r c c c c c
r * H x +E u c c c c c
(F_» G , H , E )
Represents discrete as well as interval
c’ c’ c' c transitions
Delayed Variable in Controller Memory
-f
x = u
r mm
r - x m m
(O, l. I. O)
Represents discrete transition
In this approach, the modeling work begins with the timing program. The timing program shows the switching time points during one program cycle.
Next, the state update sequence table is prepared wherein events are described as a function of time and corresponding transition equations written. Subse- quently, the overall transition matrix for one program cycle is obtained. An example is shown below in the discussion on software implementation.
To develop an overall system model of this type of system for describing its response at the program sample points t = kT, k = 0, 1, 2, . . . , two approaches are available:
• Total Transition Approach
• Incremental Transition Approach
These two approaches are briefly discussed below. Subsequently, the incre- mental transition approach is implemented as subroutine HSIMK to obtain a multirate system model with computational delays.
Total Transition Approach
The Total Transitional approach is based on the concept of finding the state response over one program period for each unit initial state vector component and for each unit input vector component. The resulting outputs form the column vectors of the total transition pair (F,G).
The use of this approach requires a certain amount of equation manipulation as discussed below.
The general form of the interconnected model is given by
T X |
= f(y, x, u) |
(244) |
y |
= g(y, x, u) |
(245) |
r |
= h(y, x, u) |
(246) |
where
x, x+ = tptal system stgrte and its update
y = collection of internal variables (internal inputs and internal outputs)
r = collection of external outputs.
69
To compute the evolution of x over one program period, the equations given above are reordered, and Equation (245) is solved for y. The result is
y |
*3 * >< 'm ii |
(247) |
+ X |
= f(y, x, u) |
(248) |
r |
ii X c |
(249) |
Now for each unit initial state vector component and for each unit input vector component, these equations are evaluated using the transition sequence table
which describes the sequence of updates on x*, x*. x^, and x^.
When all transition points within the program period are exhausted, the resulting state vector response becomes a column vector of the total transition pair (F,G).
This approach is very convenient for a paper and pencil derivation of the discrete system overall model. It bypasses a lot of matrix multiplications as required in the incremental transition approach. On the other hand, the incremental transition approach can be implemented more conveniently in software.
Incremental Transition Approach
The Incremental Transition Approach is based on the concppt of computing the total state vector and input vector, a sequence of quadruples (incremental transitions) corresponding to each transition point within the program period, and subsequently combining these to obtain the total transition over one program period.
The incremental transition approach involves three steps:
1. Calculation of the incremental transition matrices
2. Calculation of the total transition matrices
3. Simplification of the total transition matrices
The equations describing the interconnected system are in the following form (same form as single rate system):
x+ = f(y, x, u) y =g(y,x,u)
= h(y,x, u)
r
70
(250)
(251)
(252)
I riMilTMilliWMMBli
At each transition time point, the appropriate subset of Equation (250) is used with its data to obtain the matrix quadruple (AFj, AG., IL, E.) for that transition.
Subsequently, these incremental transitions are used to compute the total transition as indicated below. The total transition over one program period is in the following generic form:
x(k+l) = Fx(k) + GQ u (o) + GjU (1) + . . . Gm (r) (253)
where
u (i)
= the i-th sample of the external input within one program period
F
G.
= Total transition in one program period
= Input matrix corresponding to i-th sample of the external input in one program period.
Let
= number of time points at which u is sampled within one program period
= number of external inputs = total number of states
Now construct the n x m matrix G = [G0|G1 I . . . Gr]
where
(254)
m = nu x r
(255)
1L
It can easily be shown that the total transition pair at the i + 1 transition time point is given by
[F(i+1) | G(i+1)] = A F(i+1) [F(i) | G(i)] + [0 | AG(i+l)]
with
[F(0) G(0)] = (I | 0)
(256)
(257)
71
....... .1^^..
iiifltiiiflitti
where
A F(i+1) = incremental state transition from i to i+1 A G(i+1) - incremental input transition from i to i+1
Software Implementation
Figure 32 shows the block diagram of the state modeling software for multi- rate modeling with computational delays.
Figure 32. Flow in STAMK for Multirate System Modeling with Computational Delays
Each call to HSIMK produces the incremental quadruple (AF, AG,H, E) corres- ponding to a system transition specified by the sequence number ISQ. The total transitions are evaluated from Equation (256) from the starting sample point to the i-th transition point. When all transitions are accounted for, the output becomes the set of total transition matrices over one program period.
To facilitate the computations, state assignments are made to each hold unit
in the system (x, ) and each output variable from the controller (x ). In
11 m
cases with no delays, xm = rc and xh = xm, so that xh and xm become depen- dent variables, and the corresponding column vectors in F become zero. For this reason, the matrix quadruple as obtained above is examined before they are printed out, and the zero columns and corresponding rows are discarded from the quadruple.
t
«*■ f m* *+ ttfMk
To demonstrate the approach, we present the following example of the modeling
of a two-rate system with computational delays. Figure 33 shows the block
diagram of a two-rate system with computational delays. In this system the
inner-loop control law is executed twice as fast as the outer loop control law.
Tcl and Tc2 correspond to computational delays in each control law execution.
It is assumed that T . >T «.
cl c2
Table 6 shows the discrete system state update sequence.
C2 CX p
u T y | T+Tc2 (j T/2 iT/2+T . H
"c! > ■ "| % '*
a r 1 — 1(2 1-1 I 'i I— i_F7i — ,
Figure 33. System Block Diagram Table 6. Discrete System State Update Sequence
Sequence No. System No.
(ISQ) ISIMK (!.'<})
Updated Transition
State Interval
(kT + Tc2) (kT + Tc2)
(kT + T .)
Ci
(Tcl-Tc2»
5 |
1 |
6 |
2 |
(kT + Tcl +) (kT + T/2) (kT ♦ T/2 ♦)
(kT + T/2 + Tcl>
(T/2 - Tcl)
(kT + T/2 + T . +) c 1
(k+I )T (k+l)T +
(T/2 - Tc,)
11
3
(k+l)T ++
0
Figure 34 shows the timing program of the discrete system.
1
2
3
4
5
ucl
uc2
x
P
xc
*e2
%
j
®® ®® ©©© ®® ®o®
4— >.-l - — f-
■ ►
1 c2 '
• ; kT (V+1/2)T
(k+'l)T
Figure 34. Discrete System Timing Program
Figure 35 shows the flow diagram of the subroutine written by the user for this problem. The math-model (i.e., overall system quadruple) of the overall system at time points kT is obtained by the subprogram STAMK in the form of (FGHE).
This quadruple is the exact representation of the dynamics of this two-rate system with the delays on sample points kT. It is used in the performance evaluation program.
To demonstrate the software modeling of multivariable multirate systems with computational deiays, two specific examples are presented. Below, two-rate modeling is given for a simple system. In the Appendix C, modeling for the F-4 longitudinal system is presented for computational delays.
Example of Two-Rate Modeling By Software
The principles of the multirate modeling presented above are applied in this example using a simple system. Figure 36 (a) shows the block diagram of the continuous system: a simple lag controller, and an integral plant. Figure 36(b) shows the corresponding two-rate digital system structure. The memory unit corresponds to the digital counterpart of the hold unit.
74
1 1T'
i
i
«)' 2-RATE DIGITAL SYSTEM STRUCTURE
Figure 36. System Block Diagram
Table 7 shows the plant and the controller data. The digital controller data for this example is obtained using the z-transform for purposes of demon- stration.
Table 7. Plant and Controller Data
Continuous Data |
Discrete Data |
|
Ap = 0 |
Fp = 1 |
|
c 0) |
Bp = K |
Gp = KT |
E |
Cp = 1 |
Hp 1 |
Dp - 0 |
Ep = 0 |
|
u <u |
Ac = -1 |
-T Fc = e 1 |
0 u |
Be = 1 |
Gc = (l-e"T) |
■*■> c 0 |
Cc = 1 |
He = 1 |
U |
Dc = 0 |
Ec = 0 |
•j
i
I
i
76
The timing program of the discrete system is shown in Figure 37. From this diagram we see that xm is updated first. Next, x^ is updated. These two updates correspond to point transitions, and they take place during an arbi- trarily short time. At time T/2, xp and Xp are updated in that order. Finally, at the end of the program period (T), xc and Xp are updated. Each transition time point (point or interval transition) is assigned an interval sequence number (ISQ). This number is used for updating the states in the simulation program. Table 8 shows the discrete system update sequence.
1 2
3 4 5
6 7
(k + 1)T
Figure 37. Updating Sequence During One Program Cycle Table 8. Discrete Systejn Update Sequence
Sequence No. System No.
(ISQ) ISIMK (ISQ)
Time
Updated Transition
State Interval
kT +
kT + T
kT +T/2
kT i T/2 «•
kT + T/2 + +
(k+1 )T -
(k+l)T
77
Since two types of transitions take place in the system, we refer to this as Hybrid Simulation (Subroutine HSIMK). Figure 38 shows its flow chart. Subroutine STAMK calls HSIMK for each transition point ISQ and computes the incremental transitions (AF, AG, H, E) and the total transitions (F, G, H, E) as described in the previous section. The program documentations of HSIMK and STAMK are given in AFFDL-TR-73-119, Volume H. For this example, we can carry out the indicated transitions with paper and pencil. This yields the digital model of the two-rate system as follows:
(258)
(259)
(260)
Figure 39 shows computer results (model by software) for K = 0. 5 and T = 1 second. They agree with the analytical results computed from Equations (258), (259) and (260).
The transfer function of the two-rate system is given by
G(z) - H(zI-F)"1G + E (261)
Carrying out the indicated multiplications yields
2 z |
LJ |
,1 + e" |
T |
Ll |
-T + e |
||
2 z - |
KH -¥l |
[*- |
I=T77 |
il |
z +| |
} |
(262)
The poles and zeroes of this transfer function for K » 0. 5 and T 3 1 second agree with the poles and zeros obtained by software ( POZK). Note that (see Figure 37) using four transitions (ISQ 3 1, 2, 3, 4) and replacing T/2 by T yields the single-rate system. For this case, the analytical model is obtained as
78
QUADRUPLE OVER ONf PROGRAM P£R100
MATRIX r it- i.oooo)
1- ROW 2- ROW |
1 -COLUMN .9416327E+00 -.6321206E+R0 |
2-COLUMN .4016327E + M .3678T94E+M |
matrix 1-ROW |
FP(TP - .BOOOOE+OO) 1 -COLUMN .lmoooc+oi |
matrix 1-ROW |
FCITC - .BOOOOE+OO) 1 -COLUMN .606S30TC+00 |
MATRIX G |
(T - 1.0000) |
MATRIX |
OP (TP - .BOOOOE + OO) |
MATRIX |
QC(TC - .BOOOOE+OO) |
|
1- BOM 2- ROw |
1 -COLUMN .9836734E-01 .6321206E+00 |
1-ROW |
1 -COLUMN .2580080E+00 |
1-ROW |
1 -COLUMN .3934693E+00 |
|
matrix H |
(T - 1.0000) |
MATRIX |
HP(TP - .6C000E+00) |
MATRIX |
|HC(TC - .BOOOOE+OO) |
|
1-POM |
1 -COLUMN -.180M09E+01 |
2-COLUMN 0. |
I -ROM |
1 -COLUMN . 1M6800E + 01 |
1-ROW |
1 -COLUMN .1400M9E+91 |
matrix e |
(T - 1.0000) |
— |
MATRIX |
EP(TP - .BOOOOE+OO) |
MATRIX |
ECITC - .BOOOOE+OO) |
1 -ROM |
1 -COLUMN .I940008E+01 |
1-ROW |
1 -COLUMN 0. |
1-ROW |
1 -COLUMN o. |
Figure 39. Model by Software for K = 0. 5, T = 1 second
F(T) =
H
model.
1 |
JKT |
, G(T) = |
u |
(263) |
e"^ |
.KT)_ |
|||
M, o]. |
E = 1 |
(264) |
||
and K = 0. 5 computed results agree with this analytical |
The tr ansfer function for the single-rate model is obtained from G(z) - H(zl - F)"1 G,
80
—
and is given by:
Gj(z) =
z2 - (1 + e‘T)z + e'T
~=T,
■5 IT IT IT*
z - (1 +e A) z +e 1 + KT(l-e l)
(265)
Again, the polen and zeros of this analytical result and the computer result agree very well.
Analytical results are hard to obtain for large systems, if not impossible.
But the software approach does not suffer from this dimensionality problem. Incidentally, this example shows that G2(z) is more stable than Gj(z). The
two-rate system in this example can tolerate a 45 percent greater change in loop gain than can the single- rate system without becoming unstable.
MATHEMATICAL MODELING FOR WORD LENGTH EFFECTS
Computational errors are introduced within the digital controller due to (1) truncation of filter coefficients, (2) quantization of input data, and (3) rounding-off the results of multiplications. In the following we first develop the data truncation model. Subsequently, we consider the determination of the output noise for specified word length, and the interaction of this noise with the scaling of the control laws when fixed point arithmetic is used.
We first develop a scaling model and subsequently a digital controller noise model representing arithmetic with finite word length. Finally, we present a method for computing output noise of the digital controller as functions of scaling and word length. The details of noise analysis with fixed-point arithmetic is presented in Appendix B.
Data Truncation Model
The controller data (Fc, GC,HC,EC) are truncated or rounded to a prescribed number of bits to investigate the effects of finite data word length on controller performance. The original data (full bits) are first scaled for fractional machine representation; that is, each entry in data is expressed as
d = m 2P
(266)
where
m = mantissa of data, 1/2 < m < 1 p = exponent of data
81
Subsequently, the mantissa m is converted to a binary number and truncated to a specified number of bits. Finally, the truncated data is converted to decimal representation for performance study. Subroutine CTRUNK in the DIGIKON system performs the data truncation. It is fully documented in AFFDL-TR-73-119 Volume H.
Digital Controller Scaling Model (Dynamic Range Model)
When fixed-point arithmetic is used to evaluate the control equations in a fractional machine, computations must be scaled so that every computed number satisfies lsl<l. For safety on overflow and to avoid very detailed analysis, scaling is selected so that |s J <<1. However, to maximize the signal-to-digital noise ratio, one must select scaling so that f s | is as large as possible, subject to dynamic range constraints and transfer function invariance.
To accomplish this, we develop an Arithmetic Response Matrix, as presented below.
Structure of the Digital Controller -- The structure of the digital controller is assumed to be in the following generic form:
= Fcxc + Gcuc
(267)
= H x + E u c c c c
(268)
xft = state of controller (xx updated state) uc = input to controller r = output from controller
and (F , G . H , E ) are the controller matrix quadruples, c c c c
Form of Scalir
We divide the scaling of control laws into two groups, (a) scaling of variables (such as input, state and output) and (b) scaling of controller data (Fc, Gc,
Hc, E&). In the following we first present the scaling of variables and sub- sW# Witty the data.
TT’M '"M". ,«|» wy im?
^"ILdUU ,l |. „l
jflHP*WW» «W»T7wV''"
/
Form of Scaling for Variables — We define three diagonal scaling matrices as follows:
x = S x C X c |
(269) |
uc * Su“c |
(270) |
rc ■ Sr?c |
(271) |
where |
|
S = Scaling matrix for controller state x X c Su ■ Scaling matrix for controller input uc S = Scaling matrix for controller output r r c |
|
and |
|
x , u , r are the scaled variables c c c |
|
Substituting Equations (269), (270) and (271) into (267) and (268) yields the scaled equations |
|
— + A' _ A __ x = F x + G u c c c c c |
(272) |
•a ^ _ A _ r = H x + E u c c c c c |
(273) |
*
where
A
F S c x
G S c u
H S
C X
E S c u
(274)
(275)
(276)
(277)
Form of Scaling for Controller Data (Fc, Gc, Hc, Ec) -- The scaling of
variables as explained above transforms the original data into the form given by Equations (274) through (277). This data should now be scaled so that every element in the data is less than one in magnitude, but as large as possible.
*
83
The simplest form of data scaling is as follows:
.th
Consider Equation (272). We find the maximum element in the i row of the (FjG) pair for i=l, . . . n. Let his be s(i). Next we determine a unique the
exponent p(i) such that 2P(i) *s(i) < 2p(i). Then construct a scaling matrix
S having 2P'1' as its elements. Using this scaling matrix we write Equation (272) as
-4
x
(S * F) x + (S * G)uJ
(278)
or
x+ = S[F x+Gu]
(279)
Figure 40 shows the block diagram of this implementation.
u
Figure 40. Control Law with a Single Scale Factor Scaling Constraints
The first constraint is the invariance of "transfer characteristics" from input
to output. It can easily be shown that the structure given in Figure 41 has this property.
Figure 41. Scaled Control Equations
The second scaling constraint is called the dynamic range constraint to prevent the "overflow. " This manifests itself tc the following subconstraints: Combining Equations (272) and (273) into the following form
v = F w we write
where
• -fel
and w fc
V
vi <1. ]w. <1
Magnitude constraint:
i * 1, 2, . . .
Product constraint in the form of <1
T. w. ij 3
for all i, j
• Partial sum constraint in the form of N
iN
2>
3=1
<1
N = 2, 3, . . . i ■ 1, 2, . . .
85
(280)
(281)
(282)
(283)
Mia ii
- - ■ — *->—■**
Approach to Determining Scaling Matrices -- To determine the scaling matrices satisfying these dynamic range constraints, we define the "arithmetic response matrix" (dynamic range matrix) having the above products and partial sums (in unsealed form) as its elements. The time history of this arithmetic response matrix is then evaluated for specific inputs (step, ramp, sinusoidal, stochastic, etc. ) using simulation software TRESPK (Figures 42 and 43).
(t)
Figure 42. Arithmetic Response Matrix Time History
Scaling matrices are then selected so that every element of the arithmetic response matrix is less than one. Subsequently the scaled matrix quadruples are computed using Equations (272) and (273) and the controller software is prepared implementing the equations shown in Figure 41.
Input Considerations for Arithmetic Response — Although step, ramp and sinusoidal inputs (laboratory inputs) are used to design and test the behaviour of the controlled system, they are not too realistic for developing system response under the actual flying conditions. For this reason we chose stochastic models for generating inputs to the system.
We assume that the pilot signal is a stochastic signal with a specified rras value and bandwidth (signal generating filter). Disturbance inputs (gusts) are modeled (gust filter) similarly. The variance (a^) and the 3a value of the arithmetic response matrix are then computed using COVK. It is known that the unsealed random variables (elements of the arithmetic response matrix) will be within this range with 99. 7 percent probability (Figure 44).
Figure 44.
Bounds of Arithmetic Response r
ij
(t)
We use the 3a value of the arithmetic response covariance matrix as a bound for computing the scaling matrices Sx, and Sr#
Digital Controller Noise Model
There are four points of consideration in the control law software which determine the level and character of the round off noise for a given signal:
• The number of digits (bits) used to represent the data within the control law (i.e., Fc, Gc, Hc, l?c) and the input, output and state (i.e.. uc, rc, xc)
87
• The mode of arithmetic employed (that is fixed point or floating point), and
• The type of arithmetic (2's complement, etc. )
• The structure of the control law.
Figure 45 shows the noise model of one arithmetic cell in the evaluation of the control law: v = V w
where the partial sum ¥(i, j) satisfies
"sd, j) = ?(i, j-1) + p(i,j) (284)
and
Pd, j) = Tij w(j) (285)
The statistical properties of ^(i, j), ep (i, j) and eg(i, j) depend upon the word
length as well as the number system used in the computer and rounding or ** truncation of the lower part of the product.
Figure 45. Noise Model for One Arithmetic Cell
"ini w j a i
Ino "lSe fnalriS wittl Jolting-point arithmetic involves three steDS First SSLfijra*”.?* “oise-free-response (computation with verylowword -
nn^S r,-jfinf th? external s,*nal inputs. Subsequently, the equivldent noatinu- point noise inputs are computed as indicated above. FinaUv these are nrol^- gated using subroutine COVK. *«iauy xnese are propa-
89
1
J
.•A-'.ie.Jl-u. -V-
ip«iswpwiip*wpp"p^wmp''.w 1 "y u".
SECTION IV
SYSTEM PERFORMANCE MODELING IN STATE SPACE
Performance evaluation algorithms which are operational in Honeywell are briefly presented below. Five performance measures are considered: (1) poles and zeros, (2) frequency response, (3) RMS response to turbulence and roundoff noise, (4) power-content analysis, and (5) time response.
MODELING FOR POLES AND ZEROS (POZK)
ith
Consider the state equations describing the response of the i output to the jth input.
x a Ax + Bj Uj yi * cix + Dij uj
Transformation of this with zero initial conditions yields (si- A) X(s) - B^Ujts)
Y^s) = CjXte) + Dy Uj (S)
This set can be put in the following form:
P(s) ?(s) = qUj(s)
where
5(s) ■ col [Y^s) X(s)]
q (s) = col (D.
ij
and
P(s) =
-ci 1
IsKA),
(296)
(297)
(288)
(289)
(290)
(291)
(292)
(292)
The coefficient matrix P(s) is called the system matrix.
90
Using the Cramer's rule, we can write
Y.(s)
H-» ■ tjJtst *
det
D.,
B
1
-C.
l
sl-i
det P(s)
mi
D?s)
(294)
This is the transfer function from the input to the i**1 input.
A complex number s, is called the zero of H.,(s), if
k lj
N(sk) = 0, D(sk) * 0
Similarly, is called the pole of H^(s) if Clearly,
(295)
D(sk) = 0 N(Sk) / 0.
(296)
det P(s) = det (sI-A) = 0,
(297)
so the eigen values of A are the poles. Obtaining the zeros is more difficult. The numerator in Equation (294) can be written as
N(s) = det (Aq + AjS), where A^ is not necessarily of full rank.
For this reason, the numerator matrix is reduced to the following form:
(298)
N(s) = det
*01 |
c |
"6 |
A0+Als |
where Aj is of full rank, and Aq is an upper triangular matrix. We can now write
N(s) - K det (sI-A ), z
where
K = -detAQ detAj
A = A'1 An z 1 0
91
Therefore, the eigen values of A are the zeros of the transfer H,,(s).
Z 1]
The subroutine which implements this procedure is called POZK.
To increase the accuracy on the computed poles and zeros of a given matrix quadruple (A, B, C, D), the Newton- Raphson correction scheme may be used. Briefly, if s is the computed value of a pole or of a zero, then its improved value s is obtained from
s = s -
35 <®
(299)
The expressions for the function f(sk) and its derivative (s^) are as follows:
RESPONSE MODELING FOR REAL AND COMPLEX INPUTS
In digital control systems, some variables undergo rapid changes in real time, some variables are defined only at discrete time points, and some variables, e. g. , pitch rate and angle of attack, undergo continuous transi- tions in real time. In this type of situation, what do we mean bv "frequency response?"
Here we take the engineering point of view that we apply sinusoidal input signals to the system and measure the output under this excitation. That is, we are looking at amplitude and phase relations between continuous input/ output variables. Using this point of view, we discuss in the sequel a mathe- matical model "complex system function" which yields the amplitude and phase relations as a function of the input frequency for analog systems. Then we present the extension of this notion to systems with digital as well as analog (hybrid) elements.
92
Development of Complex Response Model
Consider a linear time-invariant continuous system described by
x = Ax + Bu (300)
y = Cx + Du (301)
If this system is stable, then the steady- state response to a complex periodic input
u = e^* (302)
can be expressed in the following form:
*(t) ■ Hx<jtt))e^ttlt (303)
y(t) = H (jui)eitt,t (304)
J
Here H^juu) and Hy(jui) sire called the "complex system functions" correspond- ing to the state and the output variables of the system. The variation (ampli- tude and phase) of Hx(juu) and Hy(jw) with respect to uu is called "the frequency response" of the system state and output, respectively.
Using the definition given by Equation (303) and the description of system given by Equations (300) and (301), we can compute the complex system function f^jiu) as follows. Differentiating Equation (303) with respect to t yields
x = Hjju)) jiu eiu,t (305)
Substituting this into Equation (300) and solving for H^juu) one obtains
Hx(jui) = (ju»I - A)*1 B (306)
Making use of Equations (302), (303), (304), and (306) yields
H (jid) = C(juul-A)"1 B + D (307)
One can find in the literature more elegant ways of deriving Equations (306) and (307). However, the concept of "complex response" introduced in Equation (303) will be of great help to us for extending the frequency response notion to digital control systems.
Digital control systems are essentially time-varying systems due to sampling operations which take place in real time. In addition, most often, the sampling operations are designed to be periodic in time, which a finite program period, Tn. Thus, the physical equations which define the evolution of response nave periodic time-varying coefficients.
93
Now we shall define the steady-state response of a digital control system to a complex input as follows [see Equation (303)]:
x (t ) = Hx(ju), t) e^1 (308)
where because of the periodicity of the complex system function
nx(ju, kTp^f^Cjuu. (k + 1) Tp], k = 0, 1, ... (309)
Here we see that the magnitude and phase of the complex system function depends not only on the input frequency, t«, but also on the time of observa- tion within the sampling period. Usually the times of observation are taken to be the sampling interval points, kTp.
Now with this restriction we find in the sequel the complex system response and corresponding frequency response for digital control systems.
By definition:
x(kT) = Hx(ja,. kT) eju,kT |
(310) |
x[(k + 1)T] = Hx [juj. (k + l)T]eju,(k+1)T |
(311) |
Hx[ju), (k + l)T] = Hx(juu. kT), k * 0, 1... |
(312) |
On the other hand, the description of system state at the sample points is given by
x[(k > 1)T] = Fx(kT) + G u(kT) (313)
where F and G are obtained by taking into considerations all transitions within the interval. Substituting Equations (310) and (311) into (313) and making use of (312) we obtain the following relation:
Hx(juu, 0)eiu,(k+1)T = F Hx(j(jj, Ote^7 + G eju,kT (314)
Simplifying this we obtain
Hx(juj, 0) = (e^I - F)_1G (315)
This is the complex system function.
Its magnitude and gain constitute the digital system frequency response observed at the sampling points. For systems with high sample rates, the time variation of the amplitude and phase response for each fixed input
94
frequency becomes small. For slowly sampled systems, the intersample "phase swing" may be auite large (in the order of 10 deg). In this case, the performance measure "phase margin" needs proper definition (i. e. , instan- taneous, max, min, average, rms, etc. ). f
;
When more than one update involving the input occurs within the sampling program period, the complex response model given in Equation (315) must be modified as described below. Figure 46 shows the input samples which are used in the control law computation of a digital control system. In this case, the overall system state, x(t), at sample points k = 0, 1, . . is described by a difference equation in the following form:
x[(k+l)T] = F x(kT)+GQu(kT) + G1u(kT+ Tl) + ... +Gmu(kT + Tm)
(316)
where F, Go, Gj . . . G^ are composite matrices which are obtained by tracing the response under the influence of these inputs over one program period.
Figure 46. Multiple Input Samples in a Program Period Now defining the complex system function as before x(kT) = H(juu, 0) u(kT)
where u(kT) is the sample value from the continuous input u(t) = e^U)t
(317)
95
(319)
ami making use of Equations (317) and (318) in (316) yields H(ju), 0) = (e^T I - F)”1 G
where
jtDTi jUJT
G=[Go + Gie 1 + * * * Gme 1 (320)
This shows that for the frequency response of multirate digital control systems. Equation (320) must be evaluated as well as Equation (319).
The Ceneral Frequency Response Software (FREQK) •
To determine the effects of sampling time on system frequency response (phase margin, gain margin), the couplex system functions defined by Equations (306) and (315) or their equivalents, as discussed below, are imple- mented in program FREQK.
Two types of data inputs are considered: 1) continuous quadruple (A, B, C,
D), and 2) discrete quadruple (F, C, H, E).
Four types of frequency response evaluations are considered. They are identified as s, d, w, and r frequency reponses as shown in Figure 47. For all types of frequency responses the transfer function is in the following generic form:
H(juu) = C[(?I-A) + in II*1 B + D (321)
In Equation (321), (A, B, C, D) matrices correspond to continuous or dis- crete system matrix quadruples. They are obtained from the simulation equation as described in Section III. The variables ? and ri depend upon the type of frequency response evaluation. Their functional relationships are given in Table 9. The complex matrix given by Equation (321) is evaluated by using the complex matrix inversion subroutine.
For a given range of frequency (number of decades), the magnitude of the elements of H(juu) are computed in units of db and phase angles in units of deg. These values are stored on permanent file for subsequent plotting. A simple plotting routine is used to see the trends in the response. Accurate plottings can be made on the "Calcomp" plotter.
96
CONTINUOUS PROCESS DIGITAL PROCESS
97
Figure 47. Frequency Response Evaluations
Variables for Various Frequency Responses
i
For increased efficiency of computation, three options are provided to evaluate (321). They are 1) direct evaluation via (321), 2) evaluation via poles and zeros, and 3) evaluation via poles and pseudo-zeros.
The option of frequency response via poles and zeros requires the poles and a set of zeros for specified input- output pairs. This data is normally avail- able (on permanent file) when a system study is made. If not available it should be generated using program POZK when this option is used.
The transfer function to be evaluated is in the following generic form:
& ( 5 -V
H( 4 ) = K
+ D
(322)
where z\, Z2 • • • zm are the zeros or pseudo-zeros of a specified input
i . i — ii i l • _ i i • m • a _ •
output pair, pi, P2 . . . pn are the poles, and K is the gain. The transmission term D is a computed quantity and its value is zero if [z^] are the zeros.
The pseudo- zeros are computed within the program FREQK if thin option is used. In the following we present a brief analysis for the pseudo-zeros of an input-output pair for a given system.
Pseudo- Zeros of an Output /Input Pair
Figure 48 shows a block diagram of a system for an output/input pair
ri' Uj
Figure 48. Block Diagram of a System for ar> Output -input I'ulr r ,
• I
99
The transfer function for this pair is given by
N,,(s)
V3’ ■ d ftr
(323)
where
N.j (s) = c. [adj (sI-A)]b. |
(324) |
and |
|
D(s) = det(sI-A) |
(325) |
We note that Equation (325) can accurately be evaluated. The direct evalua- tion of (324) should be avoided for large systems due to numerical problems. Now Equation (323) can be written as |
|
[N. ,(s) + D(s)] Hij<a> * Dtsi - + <di] - » |
(326) |
Observe that the numerator term in Equation (326) is the characteristic equation of the same system when the loop indicated by the dotted line in Figure 49 is closed. Thus |
|
fl(s) = [N^s) + D(s)] = det[sl - (A - b.c.)] |
(327) |
Hence (326) can be written as |
|
det [si - (A-b.c.)] HiJ(s) " ■ "det'(el-A) ] * * <diJ ' » |
(328) |
or n (s-z, ) |
(329) |
The zeros of N(s) are called the pseudo zeros of the r., uj pair. In summary, when the frequency response is evaluated tne pseudo zeros and poles, the poles and the pseudo zeros defined by Equations (325) and (327) are evaluated first. Subsequently, Equation (329) is used for computing the frequency response.
100
Demonstration Example for FREQK
SUbr°U‘ine FREQK "8ing * third ■"*«
x(k+l) = Fx(k) + Gu(k) y(k) = Hx(k) + E u(k)
where the system matrix quadrupled (F. G, H, E) are given as
(330)
F = |
0 0 - 0. 498047 |
1 0 -1. 88574 |
0 - 1 2. 37988 _ |
, G = |
1 1 NJ< 0 o o’ 1 1 |
H = (-0. 379882 |
0. 56152 |
-0. 18359, |
E = |
0.4 |
(331)
The sampling rate for this system is assumed to be = 25 Hz or = 157 rad/sec
(332)
syst^m.49Figure ^shoSs^e data^mace8 th.® above data into the DIG IKON 51 gives the poles and zeros correspon^^ FigUre
The corresponding transfer function is obtained as
H(z) = -~-blZ +b2Z +b3z 3 _ T, z3 + biz2 + b?z + b,
"K~ (333)
1 + ajz"1 + a2z'2 + a3z‘3
where
3 , 2
z +ajz + a2z + a3
a, =
ar
a3 =
K =
-2. 37988 1. 88574 -0. 498047 0. 4
bj = -2. 56347 b2 = 2. 44726 bg = -0. 877929
101
*
PRnr.PAM
TMAKF
rnc FIN vn.niPisq o°t = i l?/>J/n
l^o
I 0A T A I r ”"'J“:,“I^T.-»TA.TA3F^|ND|jr.T
"ssss;^:
LOr atfsamlota I'lSfQT =4w| > sr
*AWK ( 1 ) S4H»«tf MAWX (?) eAHUtl C«LL TaPp(tnsFOT,ma»<,||
*IXr1 11
N0=1
Nil: |
f<1 .|>=n.
r<i .?) = i.
F(| .D=0. f<?. n=o. f(P.?)=n. r'^D = l.
F ( 1. 1 ) = ,4V-n<, r r M.?) s-) •■'AS74 F ( 1. 0) x?, 1 7QHR
F-l t . 1 )=0.
r’<3. n=o. c» n , 1 1 = . 4
H( 1 * 1 i =-. iTQPfl?
H( ) ,?) s.SMS?
M< I • 3) *-. | J1SP
fi i • n = .4
r = .n4
CALL H°Pc<P.NX.N*,'g*.N'(.T,cMF )
call Nopf,(,0t|,3(1<T)4Hr;- ' >
CALL hp«cI-.I0.1,'J,t.4hh )
CALL ^PPMc.i.I.l.UT.AHr J
PFjnjs.joOi ihfao
rnDMflT(?nfli)
CALL TAPfJtnsFPT.HFau,!,
w ITrUl T.NX»*gp,vj,
i, ‘?,1’ -»» • T = ’ -N* » . 1 .mu, .
*» • 1=1 -NP) ,J=| ,Nx ) ,
4nr(T* J>.Is).NP»,Js|.N M
fai l TAPC ( TNSF»T,V(AP<, 1 ,
STOP
FNO
Figure 49. Quadruple Input Program
102
i'ir -ifiiTiinliTt^Vnin'iiiiHili
07100000 digital MOOf
- -J MATRIX F (T= .40000F-01)
1 -COLUMN 1-ROW 0.
P-ROW 0.
3-ROW 4.9804700F-01
MATRIX G (T= .40000F-01 ) ^
1 -COLUMN
1- ROW 0.
2- ROW 0.
3- ROW 4 , OOOOOOOF-O 1
2-COLUMN 1 •OOOOOOOE+OO
0.
-1 .«857400E*00
3-COLUMN
0.
1 .OOOOOOOF+OO 2 • 37Q8800F*00
MATRIX h ( T - .40000F-01)
1-ROW
1 -COLUMN 3. 7988200E-0 1
2-COLUMN 3-COLUMN
5*f>l SPOOOE-Ol -1 .R3S9000F-01
MATRIX E J .40000F-0])
1 -COLUMN
1-ROW 4.0 DOOE-O 1
Figure 50. Quadruple Input Image
103
V J-." J ■ "-1 J ‘ *.J "WISM/w -' .piiJin
vr»-
”.3 V“
pOlf* Of THf S7STFM
«x
MM«X a .88475994SI
7-pL*NF
PfAL |
Imat, |
oamo f>jr. |
FRl.'J |
. 747S*o<*3F *00 • 7475*00 3E *00 .884 75995E . 00 |
-.438094 S0r-m .43809ft50r-0 J 0. |
.99437484F.O0 •99*37*H4F.OO |
. 7S0?7839*".O8 . 7S077839r.no |
s |
-°t ANF |
||
Rr AL |
IMAr, |
DAMOimf, |
FRf 0 |
- . 71 8?7739E*0 1 -.7]8?7739p.oi -.3040r>730F»01 |
-.?l?87*97r «f) 1 .71 387497r*01 0. |
-.9S877946f»00 -.9S877o4(Sr.(io |
. 749J 58S7F *01 . 74915V.. 7F.01 |
7F»0S OF TR/lNSFFP FUNCTION II « I J.l = |
UMAX a
.981 ?43«499
7-pl.ANF
Rrai
.79Ul30fE*00
.79|ii308F»00
.9«1?438SE*00
Ima r,
“•*il85n7Bflr*oo .SI 850788r.n0 0.
DiMorso
.S383ft787r *00 .83436787F.00
FBFO
,945«91 PBr.QO .945891 ?flF»nn
>-°LANF
oampt^r
FPr-j
PrAL
1 390*91 0E*0 1 -.13904910E*01 ■•?731ti604E*oo
IMAr,
-. 14S04373r*0?
. !<*S043?3f*0?
0.
-.954434??f-oi
-.9S4434??f-01
. I4S7984 1 F*n? . 1457084 | F*0?
GAIN a .4000000E»00
Figure 51. Poles and Zeros
104
"*••••• * moNsrnmi i/i r« coutnct •n»ONH
•lsi or •►*! ipissci vs. o**ros
0JT*UT 1/ INPUT I
s*"pcr tint* p.oooooc-o?
iso. -I>0. -so. -*C. -30. 0. 30. 60. *0. 120. ISO. lit.
!• |
I |
f* |
1 |
!• |
I |
!• |
1 |
1 • |
1 |
I • |
1 |
1 • |
1 |
1 • |
1 |
1 • |
1 |
1 • |
1 |
1 • |
1 |
1 • |
1 |
1 • |
I |
1 • |
i |
1 • |
1 |
1 • |
t |
1 • |
1 |
I • |
1 |
1 • |
I |
1 • |
1 |
1 • |
1 |
t • |
1 |
1 • |
1 |
1 • |
f |
1 • |
1 |
1 • |
1 |
1 • |
I |
I • |
I |
1 • |
t |
1 • |
I |
1 • |
1 |
1 • |
I |
I • |
i |
• |
I |
•I |
1 |
• 1 |
t |
• | |
I |
• t |
1 |
• 1 |
I |
• 1 |
i |
• 1 |
1 |
• 1 |
1 |
• 1 |
I |
• 1 |
1 |
• T |
1 |
• I |
1 |
• 1 |
I |
•1 |
I |
1 • |
1 |
1 • |
1 |
I • |
1 |
1 • |
1 |
1 • |
I |
1 • |
I |
1 • |
I |
1 • |
I |
! • |
J |
1 • |
1 |
! • |
1 |
! • |
I |
t • |
|
I • |
1 |
I • |
T |
I • |
1 |
I* |
|
I* |
1 |
!• |
|
I* |
I |
!• |
|
!• |
1 |
!• |
I |
!• |
1 |
!• |
' r |
• |
i |
• |
|
• |
i |
so. -1*0.
-90. -60. -30.
0. 30.
90. 120. ISO. 100.
Figure 53. Plot of Phase versus jw
SSJ^fSSS? 18 Pictted ae shown In Figure. 52 and 53 using
olosel^wllh ‘wq ^uple F' G- "• E>- The "suits check very
usto Eruathion I3?3i‘ toTKf quenc,y plo‘s obuined by conventional means Y o U?ti n 3a33 * The examPle given here is for a single-input sincle-
mMiP«nityf CnV AJ* prf sented above, the Subroutine FREQK is envelope cffor multiple-input and multiple- output systems. pea 10r
NOlSEM^MCOVKf L F0R SYSTEMS WITH CONTINUOUS AND DIGITAL
?nt8eC"iVOUnd0ff n°iSe inPU*S “
First we treat the subsystem RMS responses, namely plant and controller
c.riT“oUbn"r?teyd^trc?se T™™5 re8P0"e ,0r th‘ COntinuous
RMS Response of Plant to Continuous Stationary Inputs
Consider a plant characterized by the quadruple <r G to the plants consists of two parts: p* P'
up * C0l<Upl | up2>
where
Input
(334)
upl = control input to plant up2 = disturbance input to plant
Figure 54 shows the plant block with continuous as well as sampled output
v = *(t)
:d
^x(kT)
I
I T
i
x(t)
Figure 54. Plant Block
107
(335)
The state of the plant evolved as follows
*P ■ Vp + BP1 “pi + Bp2 up2
pig^eTs)* COntr<>1 inPU, “d * etooh““<: <«»turbance
“pi'*1 * upl(kT» up2(t> = 1p <‘>
The response is given by:
kT < t s (k + l)T for all t
(336)
x(») =Fp(t-kT)x(kTHGpl(t-kT)upl(kT)+ j‘ e“-»>B ,n (.) de
kT P
(337)
Figure 55. Input Functions to Plant
gi!!eS iS a WhitC n°1Se' thC covariante response due to this input alone is X(b)=F (t-kT)X(kT) F ' (t-kT) + V (t)
r P n
where
with
v (.1 ft <t_s,Ap , (t-s)A '
Vp(t) • J e P BpWpB ' e P
kT
ds
(338)
(339)
kT < i s (k+l)T
108
and
(340)
For the stationary inputs W« is a constant matrix. In this case a change of independent variables simplifies the integral defined by (339).
Substituting
5 = t - s
in Equation (339) yields
V (t) = r(t_kT) e?AB W B' e?A,/ dt
D J P P P
(341)
(342)
At sample points we obtain
where
Xk+1 = F(T)XkF7 (T) + V <T)
VT) 1 J^^VVp <15
(343)
(344)
and T = output sample time.
The set of Equations (343) and (344) define the discrete RMS response model corresponding to continuous stochastic inputs. The intersample rms response model is given by Equations (338) and (339). In the above development, no approximation is involved. This means that the continuous covariance X(NT) obtained by integrating
i = A X + XA ' + BWB ' P P P P P
(345)
over the interval
0 < t s NT
is the same as the sampled covariance obtained by iterating Equation (343) for k = 0, 1, 2, . . . N-l
109
>W
The benefit of this model is in the saving of computing time when the plant contains high frequency dynamics. The accuracy requirement force the integration step to be too small throughout the interval
0 < t < NT
when Equation (345) is used, whereas in the discrete model onfy one sample interval
0 < t < T
small step size is needed. The steady- state values, .when they exist, are computed either from Equation (345) by substituting X ■ 0 and solving the algebraic equation, or by setting Xk+l * Xfc in (343) and solving the resulting equation. In both cases the result will be practically the same provided that F(T) and V(T) have sufficiently small errors.
This finishes the RMS response model of the plant. In the following discussions we obtain the RMS response model of the controller.
RMS Response Model for Digital Controllers with Discrete Inputs (Roundoff Noise)
The treatment of roundoff noise is given in Appendix B. Figure 56 shows the
roundoff noise model corresponding to a noise- free (ideal) controller quadruple
(F . G . H , E ). c c c c
Figure 56. Roundoff Noise Model for the Controller In this figure,
5C = Input noise vector of size n^ x 1
r) = Ouput noise vector of size n x 1 v» rc
With the unity scaling, the rms noise input values are defined as
2
Vc=E[;c(k) 5c'(k)l-
n.
"n
(346)
Wc = Eltic(k) <k>} =
m,
m.
m
m
(347)
where n^ ■ number of nonzero elements in the i-th row of (Fc Gc)
m.* munber of nonzero elements in the i-th row of (H E ) 1 c c
a « variance of roundoff noise c
The rms response of the controllers above is readily calculated from
XkH * FcXkFc +Vc
.1
\ =HcXkHc +Wc
(348)
(349)
RMS Response Model for Overall System
Figure 57 shows the overall system model corresponding to effective plant noise ?p and round off noises §c and r\c for some arbitrary system configuration.
In this model,
e{5d ? ' ) - V . Et;c ?c')-vc. E[^ T1C')=W(
and they are given by Equations (344), (346), and (347) respectively, and E{Uk uk' } = U is the command input variance matrix.
Now the problem is the development of an overall syetem covariance response model with these multiple inputs. Let us define augmented input noise ? and output noise r\ as follows:
§ = col(5p* ?c) T) = col(nc)
111
Figure 57. Overall System RMS Response Model As previously done, the overall system equations can be written as follows:
x+ = FjX + GjU^ ? |
(350) |
r. = H.x + E.u^+ n |
(351) |
ui = Pr t + Qu |
(352) |
r = Rr^ + Su |
(353) |
From Equations (351) and (352) we obtain |
|
r. = (Ir. - EjPf1 [H.x + E.Qu + t,] |
(354; |
u. = (1^ - PEj)-1 [P H.x + Qu + P n] |
(355) |
Substituting this into Equation (350) yields the overall system model in the form of |
|
x = Fx + G u + G.f + G n u V r\ 1 |
(356) |
r = Hx + E u + E_? + E n u 5 T) |
(357) |
where (F, G , H, E ) are the same as given by Equations (198) through (201) in Section Jit, and |
|
V1 VGi(Iui-piV'lp |
(358) |
V° Er, ■ R <Iri - EiP>'* |
(359) |
112
Reducing Equations (350) through (353) to (356) and (357) can also be done by software. First, an augmented input vector is defined as
u = col(u, ?, t|) = col(u, 5 » ? , r] ) (360)
P V V
of size (nu + nxp + nxc + nrc) in the w- array of SIMK. Subsequently, noise terms are added into the subsystem dynamics in the simulation equations as follows:
x* = F x + Gu + f P P P P P P |
(361) |
x + = F x + Gu + 5 c c c c c c |
(362) |
r = H x + E u P P P P P |
(363) |
r = H x + E u + t| c c c c c c |
(364) |
That is all one needs to obtain the noisy system discrete quadruple using software (STAMK). |
|
Gust Response Ratio |
If a continuous controller (i. e. , T * 0) design is based on minimizing the rms gust response, then a controller with sample time T M will produce increased rms response. We now define the rms response ratio as
Y.
l
= 20 log
10
/v>
Vrttot
(365)
where
Y. = Response ratio of the i-th output in db
R.. (T) = Variance of the i-th output corresponding to a digital 11 controller with sample time T
R„ (0) = Variance of the i-th output with continuous controller
This performance measuring stick can be used to select sample time when allowable increase is specified.
The following example demonstrates the use of the gust response ratio perfor- mance measure for sample time selection.
Consider a plant-controller combination as shown in Figure 58.
Figure 58. Continuous Control System Assume the controller is designed to create closed-loop poles of
s
1. 2
1 4. • _£L
■2*3“
The mean square value of the gust response is obtained from AX + X// + W = 0,
where
0 l" |
2 |
||
A = |
. w = |
°g 0 |
|
_- ! -1 |
-° 0 |
X = E(x x f } .
The solution is
v 2
xn " °g
X12 * -'g2/2 x22 ' “g2/2'
where og2 is the variance of the gust input.
(366)
(367)
(368)
(369)
114
Now, suppose we want to replace this controller with a digital controller as shown in Figure 59.
Figure 59. Sampled-Data Control System Three different design procedures will be considered:
1. Digitization of the continuous control law using the z- transform without hold
2. Digitization using the Tustin method
3. Direct digital design (using the same pole location criteria)
Table 10 shows the plant and controller data as functions of sample time T. Figure 60 shows the state diagram of the resulting digital control system.
Figure 60. State Diagram of the Sampled-Data System
115
Table 10. Plant and Controller Data as a Function of Sample Time T
The state response at the sample-points are obtained from
xp(k+l)
xc(k+l)
(l-T ec> |
T |
"g. |
Vk>
xc(k)
+
u(k) +
- p(k+l)T kT t^t)
0
(370)
For stationery gust input, the steady- state mean square response at sample points is obtained from
X 1 F X F7 + V, (371)
where F is the transition matrix of the above equation and
V =
(372)
where V« is the discrete equivalent of the continuous covariance W and is calculated using Equation (373).
Also,
VapVVp +w' Y°> - °- VYn
Since A = 0, we obtain V = a T. P g g
(373)
(374)
The analytical solution of Equation (371) for design procedure 1 is as follows:
where
Xll2 = P11(T) Xll(0) X222<T) = P22(T)X22(0)'
P22(T)=-
gjl+f +g T)
g T
2, -
{l-fc > - -IT <1+3fc+ecT)
i-f
’n<T>- "5
-f +g T c 6c
1+f +g T T c bc 1
A rt
99
(T).
(375)
(376)
(377)
(378)
i he solutions for design procedure 2 and 8 can also b* obtained. T«»e first component of the normalized gust response is given as
Yplf 20 log
ttr T
loyR^oT *
(379)
where
R11 = H1 X Hi (380)
The X above is defined in Equation (371), and H* is the firBt row of the output matrix H.
Equation (371) is solved using the data in Table 10 obtained by the three different design procedures. The normalized response given by Equation (379) is then evaluated for the plant output. Figure 61 shows yD versus sample time, T, for these procedures. ^
For an allowed increase of 1. 5 db 'the Tustin controller requires T = 1/2- second sample time.
118
^Word- Length Roundoff Noise Relatinna
tTradefoff7roMemVelThe^ ‘Cfdlffr apprSac.h t0 thc sample-rate-word-length
ssrzv«“ sr^sr *«-
Consider » continuous controller with dynamics described by
x = A X + B u c c
Let the sample time be T seconds. Substituting
x =
Xk+1 ~ xk
into (381) yields
where
and
xkH "Fc(T,xk+Gc(T)uk
F„(T) = (I + A T)
v- C
G (T) = B T c c
(381)
(382)
(383)
(384)
(385)
Equation (382) is the "first-difference" algorithm. Using Appendix B the roundoff noise model is given by Figure 62.
Figure 62. Roundoff Noise Model of the Digitized Controller Dynamics
119
(386)
*k+l = Fc(T)3?k + ^w
where t) is the roundoff noise vector with the variance matrix given as
W
12
I
W (w) = where w s word length.
The steady-state value of the mean- square error is given by
X = F (T)X F' (T) + W c c c c
Substituting (384) into (388) yields the following matrix equation XcAc' + AcXc + AcXcAoT + W ■ 0
(387)
(388)
(389)
This is the functional relation between the noise covariance, continuous system dynamics, sample time, and word-length parameters.
As an example consider the following first order differential equation:
Let
x * a x + b u c c
*k+l * fcxk + 6c“k
(390)
(391)
be its discrete representation. Let a be the roundoff noise variance in the computation of the right-hand sidewof (391) and a * be the resulting output noise variance. x
The steady-state solution to Equation (°°9) exists when:
«fc< 0 (392)
and
0 < | ac j T < 2 (393)
Then the use of Equation (389) yields
Hsel + M*T)"«,+-
W
(394)
120
Defining the digital noise amplification factor as
a
IT*'
W
(395)
and solving Equation (394) for T yields
t=t u
c >-
1
V V
(396)
This shows that for a fixed noise amplification level, the sample time is inversely proportioned to the pole location. Smaller pole locations Require higher sampling times (lower sampling rate).
Noting 'that
w
2 (2"w)2 = 12
(397)
where w is the word length, another form of solution of Equation (394) is given by
w = log 2
I, 2 vy /f v2 |^| . a^T
(398)
This shows that for a given output noise level, lower sample times require longer word lengths. The smaller the pole, the higher the required word length. The third form of the solution of Equation (396) is given by
1
22w (12) T (2 [a_ - a2T)
c c
(399)
This indicates that in order to keep the digital output noise variance down, word length and sample time must be increased. The smaller the pole location, the more dominant its contribution is to the output digital noise.
POWER AND POWER SPECTRAL DENSITY MODELING FOR FREQUENCY TRUNCATION (POWK)
For signals generated in physical systems, the power content of a signal in a prescribed frequency band can be used to determine significant frequencies of the signal in that band. This, in. turn, can be used to indicate how fast the sampling rate should be so that the digital signal is transmitted through the discrete channel without a significant loss of signal power. In this paragraph we develop the power content model. In Section V this is applied to a simple system.
121
I
t
l
I
I
TT
Figure 63 shows three major stochastic inputs to a control system:
• Signal input s(t)
• Gust input Wg(t)
• Sensor noise input n(t)
These inputs are assumed to be generated by the corresponding filters having independent white-noise inputs , and shown in Figure 63.
The total output spectral density is obtained from
where
n
S.(ou) = I 1 k = 1
Hik(jui)
Sk(uu)
(400)
Sk(ui) = power spectral density of input
magnitude < output, and
H^jw)
magnitude of frequency response from k^1 input to i*
S.(uu) = power spectral density of output yi
n
Figure 63. Stochastic Inputs to a Control System The signal (or noise) power lying in the band 0 i u> s uu0 i8 obtained from
w .
P(s) = f° ±S(uu)dui.
w TT
(401)
122
The steady- state power level (mean- square value of signal) can be obtained from
uu ,
jtim I* ° — S(uu) duu.
<J TT
® — O
(402)
Equation (401) shows the power content of the signal in the band 0 s w s uQ. This can be used to determine the significant frequencies of the signal by effectively truncating the frequencies (theoretically the frequencies go to in finity but practically they are insignificant beyond some power settling fre- quency). This fact is illustrated in Figure 64.
Figure 64. Power Spectral Density and Power as a Function of cj
The power level is said to be settled when it reaches p percent of its steady state value (for example when p is between 90 to 95 percent of its steady- state value). The corresponding bandwidth is called the power settling fre- quency or the settling bandwidth. (This is analogous to the 50 percent power point for the regular bandwidth definition. )
To obtain the "settling bandwidth of a signal", normalized power is computed. The normalization factor is the steady- state power level (mean- squared value) of the signal. It is obtained by solving the following equation for continuous signals:
X = 0 = AX + XA* + BWB
Y = CXC ' + DWD /
where W is the disturbance covariance matrix.
(403)
For any given sampling frequency, the total average signal power of the digital system is computed from
P(id) = — f8 I H* (eju)t)
d)„ •*
S(uu) duu
s o
(404)
123
where
S(uu) = Spectral density of the digital input signal,
lH*(e^mtJ= Digital-frequency response amplitude from input to output,
uu = Sampling frequency (rad/ sec) and
s
P(uu) = Average power content of digital output signal.
For digital signals, the following equation is solved for the steady- state power density levels
X = FXF /+ GWdG/
Y = HXH/ + EW.e'. where Wd = W/T. d
(405)
Then the densities are integrated in the frequency domain until the powers reach their settling levels.
Program POWK implementes this analysis. It is fully documented in Volume II of this report. A demonstration example is given in Section V of this report using a fourth-order system model.
TIME RESPONSE MODEL FOR DETERMINISTIC INPUTS (TRESPK)
The second order algorithm [8], given below, is used in integrating the differential equations to get the states and responses to deterministic inputs
Vri*T (3ik- Vi1
where AT is the integration step size.
The derivatives are either computed directly using the matrix quadruple ABCD in
x = Ax + Bu
or are obtained from the simulation equations [see Equations (4) and (5)]. Since in this case x appears in both sides of these equations, the aged deriva- tive xk_j is used to compute the current derivative x^.
In the discrete case the states and responses are merely updated using the digital quadruple FGHE in
Vi = F \ + Guk
rk ' Hxk+i + Euk+i
These expressions are implemented in time response program TRESPK for step inputs and fully documented in Volume II, Section VIII.
124
SECTION V
COMPUTATIONAL REQUIREMENTS AND PARAMETRIC STUDY
This section documents a comprehensive study of digital flight control param- eters. Aircraft flight condition, system bandwidth, sample-rate, and word length are to be varied, and the relative influence on performance is to be examined. The objective here is to define computation rate requirements for a tactical fighter and the rate sensitivity to DFCS parameters.
The F-4 longitudinal control system presented in the fly-by-wire report AFFDL-TR-71-20, Supplement 2, was selected for the parametric study, which was carried out in two levels of system complexity. First, the F-4 longitudinal structural filter was investigated. Subsequently, the overall F-4 longitudinal control system (open loop and closed loop) was studied. These studies are summarized in that order.
The various topics discussed in this section are supported by numerous figures. To preserve reader continuity, therefore, each topic will be presented in its entirety and then followed by its supporting figures. However, there are a few obvious exceptions to this format where small figures are presented within the text.
125
if i rii~M6ai'-ia*iaaiay
PARAMETRIC STUDY OF A STRUCTURAL FILTER IN THE F-4 LONGITUDINAL CONTROL SYSTEM
Parametric analysis by software was carried out to relate the poles and zeros and the frequency response of a structural filter to the computational param- eters--sample time, and the coefficient word length. The structural filter is the same as that used in the F-4 longitudinal control system.
The following parameter set was used:
• Sample Time: 0,1/1000, 1/160, 1/80, 1/40, 1/20 sec
• Coefficient Word Length: 24, 16, 12, 8 bits
Figure 66 shows the transfer function, state diagram, and differential equa- tions which describe the dynamics of the structural filter (which is also called a notch filter), and Figure 67 shows the program listing describing the contin- uous filter in Subroutine SINKC. Figures 68 and 69 show the sample-time root locus in the image s-plane and z-plane of the notch filter based on the pole -zero data for a 16 -bit coefficient word length. The zeros are computed for sampled-output/sampled-input transfer. Figures 71 through 75 (presen- ted following this discussion) show the filter quadruple and associated poles and zeros for a 16-bit coefficient wordlength and sample times of 1/1000,
1/160, 1/80, 1/40, and 1/20 sec., respectively. Figure 70 with T =0, Full Word is included for comparison. Figure 76 shows the effect of coefficient word length on the quadruple data. For sample time T = 1/80 sec, 24- and 8-bit data are displayed. Figures 79 through 86 show the dependence of the frequency response (gain vs. omega, and phase vs. omega) to the sample time parameter for a fixed word length. This dependence is exhibited for 16 bits of data and sample times of 1/160, 1/80, 1/40, and 1/20 seconds respec- tively, using sampled and zero-order-held input and zero-order-hold output. Figures 77 and 79 (T = 0, Full Bits) are shown for comparison purposes.
Figures 87 through 90 show the dependence of the frequency response (gain vs. omega) to coefficient word length for fixed sample time. This dependence is exhibited for a fixed sample time of 1/1000 sec. , and word lengths of 24, 16,
12 and 8 bits, respectively, using sampled and zero-order<ihold input and * i zero-order-hold output. Figure 91 shows the frequency response table. Fig- ure 92 shows the loss of phase margin.
We note that in this part of the parameteric study, we used a subsystem ap- proach (a short cut) to sample rate selection. In this approach, a critical subsystem is chosen and isolated from the rest of the system. Subsequently its variation (i. e. , deterioration) from the ideal is investigated as a function of sample time and word length. Maximum allowable variation determines the computational parameters.
Figure 65 shows replacement of a continuous controller by a digital controller in a feedback system between the terminals A and B. Within the controller, an element which is most sensitive to sample rate is the structural filter.
126
' ")"’1! '■ 1 m*r
Figure 65. Replacement of Continuous Controller with a Digital Controller in a Feedback System
The following conclusions can be drawn from analysis of the parametric studies:
• A coefficient word length of 16 -bits is sufficient to represent the discrete notch filter dynamics (i.e., difference equations)
• The sample-time root locus in the image s-plane shows that the notch frequency and damping is very sensitive to sample time. They are both reduced by increased sample time. The complex poles of the filter have the same trend.
The roll-off filter bandwidth increases 30 percent when sample time is increased from zero to T = 1/80 sec. This shows that for sample times greater than 1/80 sec. , poles and zeros must be prewarped to maintain critical frequencies.
• Frequem^ response plots show the notch frequency shift to the lower frequencies as sample time is increased from zero. High frequen- cies are sharply attenuated due to a zero introduced by the Tustin algorithm at the half sampling frequency, and, due to the attenuation characteristics of a zero-order hold unit. This attenuation, how- ever, is obtained with an excessive phase lag (approximately 90 deg at half sample frequency) as shown in Figures 87 through 90.
• If the additional phase lag introduced by the digitization of the filter and by the hold unit is to be constrained to some maximum value at some critical frequency, then the sample rate can be chosen accordingly.
• Figure 92 shows that at w = 10. 25 rad/sec (approximately the air- craft rigid body crossover frequency), a loss of 3 degrees in phase margin corresponds to sample time of T = 1/100 sec.
127
DIFFERENTIAL EQUATIONS
V1’ = Uf Yc(1)
X (2) = X (3) c c
Xc<3) = Ud Yc<2>
;n lud}2
Yc(l) = -XC<1> + Xc(2) + 2 jj- Xc<3) + |~ Yc(2)
id " "
Y (2) = U(l> - 2 — X <3> - X (2) c c c
r(l) = Xc(l)
OV ' i -o«c( 1 otfco! ! uuo
SUBROUTINE SIMKC
j
> THfRO ORDER NOTCH hlTER DYNAMICS
« *
COMMON V(4l)*W(?0)«NX*NY«NR*NU«INITt I SO* MODE «F (41*70)* TPS» IFLAi.T 0IMEN5TW xOgTH) » X ( 3) « Dm ...
TXCOTTn. W (1 ) ) • (YU). W(4)). TXU)» W(6))»
4 f«U)» «(9I)
ir UNIT .NE. 0) SO TO 100 NX ■ 3 NP = 1
WT5 I
NY * 2
RETURN 1ft CONTINUE NN * 86.
“ XlN = .05
"TO
W* • 120.
MOOT EQUATIONS
VTTT = wr ^ vm VT2T * X (3)
¥<31 « Wp * MO * Y<2)
T EQUATIONS
VT5T = -XTT) ♦ XT2) ♦ ((?. * XTN) / WN) * X C3) ♦
I TTWa* WT) / IWN * WN)) * V f 2) )
VtS) « U(l) - < (2. • X ID / WO) * X(3>>- X ( 2)
RESPONSE EQUATIONS
VT51 = Xfl)
RETURN
END
Figure 67. Notch Filter Simulation Program Listing
129
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Figure 69. Notch Filter Sample -Time Root Locus in the z- Plane
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Figure 81. Filter Gain (db) versus Omega for Sample Time T = 1/80 sec and Word Length = 16 Bits
Figure 82. Filter Phase (deg) versus Omega for Sample Ti T = 1/80 sec and Word Length = 16 Bits
Figure 83. Filter Gain (db) versus Omega for Sample Time T = 1/40 sec and Word Length = 16 Bits
Figure 84. Filter Phase (deg) versus Omega for Sample Time T = 1/40 sec and Word Length = 16 Bits
Figure 85. Filter Gain (db) versus Omega for Sample Time T = 1/20 sec and Word Length = 16 Bits
I
Figure 86. Filter Phase (deg) versus Omega for Sample Time T = 1/20 sec and Word Length = 16 Bits
Figure 87. Filter Gain (db) versus Omega for Word Length = 24 Bits and Sample Time T = 1/1000 sec
Figure 88. Filter Gain (db) versus Omega for Word Length = 16 Bits and Sample Time T = 1/1000 sec
Figure 89. Filter Gain (db) versus Omega for Word Length = 12 Bits and Sample Time T = 1/1000 sec
Figure 90. Filter Gain (db) versus Omega for Word Length = 8 Bits and Sample Time T = 1/1000 sec
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Figure 91. Frequency Response Table, Omega versus Sample Time (Continued)
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POWER CONTENT ANALYSIS WITH STRUCTURAL FILTER
Figure 93 illustrates the fourth-order system transfer function model involved in this study.
(Signal Filter) (Structural Filter)
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The first block represents the input-generating filter. The second block represents the analog process, which consists of a notch filter cascaded to a roll-off filter.
The parameter values are given in Table 11.
Table 11. Parameter Values
Filter |
Parameters |
SigrfcSt |
= 200, .2 rad/sec |
Processor |
-Tjr- = 86 rad/sec, Cn = . 05 n ijr- = 84 rad/sec, Cd = .6 i- = 120 rad/sec xf |
Figure 94 shows the system quadruple with the 200 rad/ sec prefilter. Figure 95 lists the power and the power- spectral density (PSD) as a function of omega, and Figure 96 plots the same. Figure 97 shows the plot of the power and the density with 0.2 rad/ sec prefilter. It is seen from Figure 96 that approxi- mately 95 percent of the signal power is in the band of 0 < u <u_af+, where usett = 325 rad/ sec. u
172
Un the basis of the settling frequency of the continuous process, the Nyquist frequency (reflection-frequency) of the digital process is computed as
id = k ui .. where, k * 1.
nq sett
The required sample time is then given by
T = 'v ■)TT second
K sett
For the above process, and assuming k ■ 1, one obtains T * 0. 01 second as a required sample time.
In this approach, the sample time T is dependent on the power- settling fre- quency of the output. This, in turn, depends on the processor as well as the input signal spectral content. For instance, if the signal-generating filter band width is 0. 2 rad/sec, the 95 percent of the output signal power lies in the band of 0 <uu<ousett* where uugett = 2. 5 rad/sec. (See Figure 37.'
Thus, the corresponding sample time would be (for k = 1)
T = = 1.25 seconds.
In summary, this brief analysis shows that the upper bound on the sample time requirement is obtained by assuming a white noise process as a signal process. In this case, the settling frequency of the filter alone (open loop) gives an upper bound on the sample time.
For a colored signal generated by a 200 rad/sec prefilter, 95 percent of total output power lies in 0 * ids 325 rad/sec. bandwidth. Truncating the frequency response at this frequency and applying the sampling theorem to this truncated process, one obtains T = 0. 01 second as a required sample time.
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176
PARAMETRIC STUDY OF F-4 LONGITUDINAL CONTROL SYSTEM
The parametric study of F-4 longitudinal control system is presented in this section.
First, a brief presentation of modeling efforts is given. This is followed by a parametric study of F-4 longitudinal eontrol system stability and frequency response performance. The "first quadrant rule” for sample rate selection is developed in this section. Also, conclusions are listed with respect to requirements on the bafcis of stability and performance measures.
Subsequent paragraphs present a parametric study of F-4 longitudinal control system gust response ratio performance and conclusions drawn for the require- ments based on this performance measure. This is followed by a parametric study of F-4 control system stability with computational time delay, and conclusions drawn from this study.
Modeling of F-4 Longitudinal Control System
The F-4 longitudinal block diagram (vehicle, sensor dynamics, actuator dynamics and controller) presented in the fly-by-wire report AFFDL-TR-71-20, supplement 2, is used to generate the system model by the DIGIKON software. The FC-11 with Mach 1. 2, 5000-ft. flight condition (q max) is chosen because of model frequency considerations (highest aeroelastic frequencies). Three bending modes are included in the aircraft model. Figure 98 shows the four subsystem blocks that comprise the overall system and the interconnection between the blocks.
The procedure for generating models by the DIGIKON software is briefly outlined as follows: Starting with the physical equations or the system block diagram, a simulation diagram is shown. From the simulation diagram, the state equations, 3umming point equations, and response equations are written. These equations are then programmed for the DIGIKON software. A similar procedure is followed for the controller, sensors, and actuators. After the subsystems have been verified, they are interconnected as shown in Figure 98.
In the following, first models (i. e. , quadruples) for the subsystems are obtained. Subsequently, the actuator, vehicle and sensor subsystems are combined into one system called the plant. Finally, the plant and controller are combined into the overall system.
Figure 99 shows the block diagram for the sensors. The state diagram is presented in Figure 100 and the sensor equations (differential equations, summing point equations, response equations) are given in Figure 101. Figure 102 is the program listing of subroutine SIMKS which implements the sensor equations.
178
m
Figure 103 shows the physical equations for the vehicle (A/C), The simulation diagram is presented in Figure H)4, and the vehicle equations (differential equations, summing point equations, response equations) are given in Figure 105. The FC-data is listed in Figure 106. Figure 107 is the program listing of subroutine SIMKV which implements the vehicle equations.
Figure 108 shows the block diagram for the actuator. The state diagram is presented in Figure 109 and actuator equations (differential equations, summing point equations, response equations) are given in Figure 110. Figure 111 is the program listing of subroutine SIMKA which implements the actuator equations.
Figure 112 shows the block diagram for the controller. The state diagram is presented in Figure 113 and controller equations (differential equations, summing point equations, response equations) are given in Figure 114. Figure 115 is the program listing of subroutine SI3VTKC which implements the controller equa- tions. Appendix A documents the controller modeling via transfer function input. This approach to modeling is more convenient when subsystems are described by transfer functions.
Figure 116 shows the block diagram for the plant (A + V + S). The plant equa- tions (differential equations, output equations, interconnection equations, plant outputs) are given in Figure 117. Figure 118 is the program listing of subroutine SIMKP which implements these equations.
n -
Figure 119 shows the block diagram of the overall system (P + C). The overall system equations (differential equations, output equation^, interconnection equations, overall system outputs) are given in Figure. 120. Figiye 121 represents the program listing of subroutine SIMK which implements these equations.
When the mode switch is closed (MODE = 0) the overall closed loop system model is developed. When the mode switch is open (MODE = 1) the overall open loop system is obtained.
We note that the modeling through subsystems as described above provides modularity. This facilitates subsystem modification, and checkout. It goes without saying that the actuator, vehicle, and sensor groups can be modeled together as one subsystem. The user chooses these options for his needsT
179
1
Figure 98. F-4 Simulation Interconnection
Figure 99. Sensor Block Diagrams
Figure 100. Sensor State Diagrams
8
NAME LIST FOR SENSOR
XS(1)=PITCH RATE GYRO STATE 1 XS(2)=PITCH RATE GYRO STATE 2 XS(3)=NORMAL ACCELEROMETER STATE RS(1)=PITCH RATE GYRO OUTPUT RS(2)=NCRMAL ACCELEROMETER OUTPUT US(1)=PITCH RATE GYRO INPUT US(2)=NORMAL ACCELEROMETER INPUT
Differential Equations
\(1) = "a *S{1) is(2) = xs<3)
is(3) = aiq2 ys(2)
Summing Point Equations
ysU) - Sa us;,:, - Xs(l) yS(2) ■ Cd/r Bq \(1) ' x8(2) ‘
xs(3)
Response Equations
rs(D = Xfl(2) ra(2) = xs(l)
Values of the Parameters
CD/B ™ 57- 29578
V *12
U)q = 150
Ba = .8
id = 200 a
Figure 101. Sensor Equations
i
;
182
I
SUrPOUTINE SINKS CDC 6600 ETN V3, 0-7355 OPT-1 11/13/73 00.67.37,
5
10
15
V
20
25
C
SURROUTINE SINKS SIMKS 6600 VERSION
NX«3
NR«2
NU*2
NY«2
RETURN
100 CONTINUE WA-200. BETAA».8 WQs 1 50 . BETAQ*.12 XIQ*.6 CDPR.57.3 V < 1 ) aWA*Y < 1 )
V <2) *X (3)
V (3) >WO#wQ*Y (2)
C Y EQUATIONS
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c»tspoVM ( 1 * -* ' * * - ' < 2- ** ,»> '-a* ** • 3 >
V(6)»X(2)
V ( 7) «X ( 1 )
RETURN
end
Figure 102. Program Listing for Sensors
Figure 104. Vehicle Simulation Diagram
185
NAME LIST FOR VEHICLE
XV(1)=ANGLE OF ATTACK (ALPHA, RAD)
XV(2)=PITCH RATE (Q, RAD, SEC)
XV(3)=STABILATOR BENDING (ETA1)
XV(4)=STABILATOR BENDING RATE (ETA 1 DOT)
XV(5)=FIRST VERTICAL BENDING (ETA2)
XV(6)=FIRST VERTICAL BENDING RATE (ETA2DOT) XV(7)=STABILATOR ROTATION (ETA3)
XV(8)=STABILATOR ROTATION RATE (ETA3DOT)
XV(9)=GUST ANGLE OF ATTACK (ALPHA, RAD)
RV(l)*TOTAL PITCH RATE AT GYRO LOCATION (QT, RAD/SEC) RV(2)=TOTAL NORMAL ACCELERATION AT ACCELEROMETER LOCA (NA IN/SEC)
UV(l)=STABILATOR DEFLECTION INPUT (DELTA, RAD)
UV ( 2 ) =ST A BILAT OR RATE INPUT (DELTADOT, RAD/SEC) UV(3)=STABIT,ATOR ACCELERATION INPUT (DELTADDOT, RAD/SEC2) UV(4)=WHITE NOISE INPUT TO GUST FILTER
Differential Equations
Xv(l) - (1 + z-) Xy(2) + Za Yv(l) + Z • Xv<4>
+ Z^ Xv(3) + Z^ Xy(6 ) + Z^ V5' + ^ V8>
+ Z Xy(7) + Z'j Uy(3) + Zj Uy(2) + Z, Uy(l)
■ M<; V2) + “aV11 + Me xv(2) + % V4)
+ Xy(3)+Mfi Xv(«> + Mr) Xy(5) + Xy(e)
1 2 2 3
+ M^ Xy(7) + Mg Uy<3) + M^ Uv(2) + Mfi Uy(l)
3
Xv<3) = Xy(4)
V4) = Fa V,|tFiV21 + F?) V * * * * Xv(4) + Fti Xv(3) + Fi Xv(G>
112
+ F Xv<5) + F- Xv(8) + F Xy(7) + F'j Uy(3)
& u o
+ F6 Uv(2) + F6 Uv(1)
Figure 105. Vehicle Equations
186
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112
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+ GjUy(2)+GsUy(l)
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V V
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11m
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2 3 3
+ Hj Uy(2) + Hs Uy (1)
xy<9) - t- [- xy(9) + ^jrrw (a/00> Uy(4)j
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Yy(2) . Xy(l)
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V4> ■ TTT- [«v(2> - Xv(13
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V6> * V5> 'TI7T [«1 Xv(4> + h Xv(6> + <3 K (8)]
Response Equations ry(l) = Yy(l) ry(2) = Yy<6>
Figure 105. Vehicle Equations (Concluded)
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S (fi O iSDCSSlffiirAtDS (S O (O A (D 0 (S 0 0 -J-*_*_J_J_J-J_J -J J J J J J J J J J J J JJ M OA. C-«V OCV.C M O M O M C * M t ftjOft/OM/ w ft, w m r (\j w ft w r. r*> ft w ftjwi/vjnftjnftir'ft, S KKKKKKKKKKNKKNKKKNKK .V fs
10
^vvx m/yvYi/in/wiryirMviP^vv uiu'ininminintnintninintninintnininininin *00
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iPiPtPlPffKMMMMPfti/VilOiriPl/'iin^ili *> — —
HftiW <PO KDffi OMftr
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HH^^HHftftft
Figure 106. Flight Condition Data (Concluded)
SUBROUTINE
SM'V
CDC 6600 FTN V3.0-P35S OPT*l 11/13/73
00.47.37
5
10
IS
20
2S
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IS
*0
*5
so
55
60
65
70
SUBROUTINE S|Mrv f
c simulation tiu.TiONS fq» f-4vemicle
C SImkV 6600 V-’BSION
c
COMMON V(4| I ,W(701.NX.NT,NR.NU.IN1T,IFlAG. NODE. DUKE 141.701 *T* If c COMMON/OTA-'E/MARS (20 1* locate * insert, null OImenSIOn /(lll.N(i2i,F(ni.G(ll).M(ll)
DIMENSION (D0T(9).V(6).X(9).U(4).R'2I DIMENSION iRUF(68l
EQUIVALENCE (XDOTUi.WIIi l.(T(l| «W(I9)>. (Oil)* Wll6ll.IUIll.HI ESI I EQUIVALENCE IAAuF (5) .11 1 1 I . (ABUFI16I .Mil I I , (ABUT (Eft .Ml) ) .
I (A8JFI39I . G < 1 > 1 1 (ABuFISOI.mUI I , < A«Uf (6 1 1 . 0*M III .
» | APUF (ft?) * DPM 1 2 1 . ( AAUF (631 • DPMI 3 1 . ( ABUF (44I.PHIII.
1 (AHUf (SSI .PHI?t,IASuri66).BHI]l.(AaUM6T).LENflTHI
k. ( ABUF (6“1 .UPS? I BEAL N.LX.ifNGTw If (INIT.NE.0I GO TO 100 C
C INITIALIZE
c
CALL OATAIidUf.il NX > <9 NT » 6 N» » 2
NU • 6
LX » <l£ 'CiTH - 77.01 / 12,0 C
C MINI", FILTER IN’I/T
c
T» ■ 0.072*. sir,** a • 70 ,o
RETURN
100 CONTINUE C
C OIFFERfNTIAL E UATionS C
V(|l • 7I|1»yU1 • (I.O.Z(2M*X(2l • 7I31*XI4I . Z(4|»X(3I !♦ 7(SI»X(6I • 7 (61 *x (S) • Z(TI*X(8I . 7 ( A I *X 1 7 1 ♦ Z(9I*U(3>
’• 7 ( 1 0 1 *u( * i . 7(ll)*U(ll
V ( 7 1 » ■ ( 1 I *7 ( | 1 • M(2)«Y(Zl » M ( 31 #X ( 2 I . N (6) *X (4 1
I » mcsioxOi . m(6I»X(6> • N(T)»X(5I • M(BI*X(*I ♦ M(91*K(7|
7« m(|2i«i(1I * M(lll*Uf?l • M 1 1 0 1 *U ( 1 1
VIII « . (1.1
v(*i • e ( 1 1 « f ( i ) . r (2l*x (2) • f ( 31 *x ( 4 1 ♦ F (41 »x ( 31 I ♦ F(SI«X(6I « F (61 *x (SI ♦ F 1 7 1 • X (8 1 • F 1 8 1 *X 1 7 1 » F(9I*UI3)
3. F(lOl»U|E| . f I 1 1 1 *U ( 1 1 V(SI • < 161
V ( 4 ) • '(1I«T(|I . G( 21 *x < 2 1 ♦ G 1 3< *X (61 • G(4)«X(3I
I* e.(SI»X(6i • f, ( 6 1 • x ( 5 1 • G(7|»X(8I • r,(8l*X(TI ♦ G(9I*U(3>
■>. r.(10|*.i(?i . otl)i*UU>
V ( 7 1 < (81
V ( A I » ->(11*7(11 ♦ H ( 21 •» (2> » H I 3 1 • X ( 4 I » H <41 *X ( 3 I I • N(S1*X(4I • M(61»X(5» ♦ H(T»»Xt81 « M(BI*X(7) • M(9I*U(3I
»• “U5j*jji5> • iiuiumi
V < 9 1 » ("X (9) . SORT (2.0*TWI*SIGMA*U(4I /UPSZ I /TW
C
C COmouTf y EQUATIONS
c
(Mini > .191 . XIII Vdll « «D Till
/IIP! * «(2I . 0°M|1»X(4I ♦ OPMI2»X(6» ♦ 0PhI3*X(8I V ( 1 3 1 • (UOSZ/32.2l*(X(2)-X00T(l) I
V ( 1* I * 7(41 . (LX/32. 2I*XD0T(2)
Vd il' « v(5l - (»MIl«X00T(4).pMI2»XD0TI61*PMI3*XD0T(#ll/32.2
C
c PATF AND ACCELERATION OUTPUT C
V ( 16) • 7(3)
V ( 1 7 | >7(61
RETURN ENO
Figure 107. Program Listing for Vehicle Equations
192
■waiiaiitifii
8. Actuator Block Diagram
NAME LIST FOR ACTUATOR
XA(l)=POWER ACTUATOR STATE (DELTAE, RAD) XA(2)=SECONDARY ACTUATOR INTEGRATOR STATE XA(3)=SECONDA RY ACTUATOR (SERVO VALVE) STATE XA(4)=SERVO AMPLIFIER STATE
XA(5)=DEMODULATOR (FILTER) STATE (SECONDARY ACTUATOR) XA(6)- WASHOUT FILTER STATE (SECONDARY ACTUATOR) XA(7)=DEMODULATOR (FILTER) STATE (STABILATOR ACTUATOR) XA(8)= WASHOUT FILTER STATE (STABILATOR ACTUATOR) XA(9)=ACTUATOR INPUT FILTER STATE RA(1)=STAB1LAT0R DEFLECTION (RAD)
RA(2)=STABILATOR DEFLECTION RATE (RAD/SEC) RA(3)=STABILATOR DEFLECTION ACCELERATION (RAD/SEC2) UA(l)=ELEVATOR CONTROL INPUT TO ACTUATOR
Differential Equations
x(l) = KMR * Y(l) x(2) = x(3) x(3) = W3 * Y (2) x(4) = W2 * Y (3) x(5) = WDMOD1 * Y(6) x(6) = Y(5)
x(7) = WDMOD2 * Y(6)
x(8) = Y (7)
x(9) = W1 * Y(9)
Summing Point Equations
Y(l) = LINKG1 * X(2) - CSTABA * Y.i 1) Y(2) = BETA 3 * X(4) - X(3)
Y(3) = BET A 2 Y(8) - X(4)
Y(4) = -X(5) + CVPIN * BDMOD1 * X(2)
Y (5) = -X(6) + X(5)
Y(6) = -X(7) + BDMOD2 * X(l)
Y (7) = -X(8) + X(7)
Y(8) = X(9) - CGOK * Y(5)
Y(9) = BETA1 * Y(10) - X(9)
Y(10) = U(l) - Y(7)
Figure 1 10. Actuator Equations
195
Response Equations
r(l) ■ LINKG2 * CRPD * X(l) r(2) = LINKG2 * CRPD * X(l)
r(3) = KMR * LINKG2 * CRPD * (LINKGL * x(2) - CSTABA * x(l))
Values of the Parameters
BETA1 = .37 BETA2 = 57.6 BETA3 = .408 BDMOD1 = 1.25 BDMOD2 « 1.17 LINKG1 = 1.372 LINKG2 = 2.8G5 KMR = 163.
CSTABA = 1./7.128 CRPD = 1. /57.3 CVPIN = 14.
CGOK = .296 W1 = 1000.
W2 = 1500.
W3 = 565.
WDMOD1 = 1000.
WDMOD2 = 200.
Figure 110. Actuator Equations (Concluded)
196
SUrROUT INF SIM*A CCC 6600 FTN Y3.0-P355 OPT.| 11/12/73 23.27.23.
5
10
15
> 0
25
30
35
60
65
50
.55.
SUBROUT I ME SIMka C SIR«* 6600 VISION
C S 1 Mij_ AT I ON E1U«TiONS FOP F-6 ACTUATOR C
COMMON V(6t ) «W ( 70) .NX .NY.NR.NU. INI T« IFLAG.MOOE.F <61 .','0) .T.IFC DIMENSION «<9>.XD0T<9> ,T(10| iUII )
REAL KMR.LINKG1.L1NKG2
eO'Mvalencp (x dot ( i > .*( i> > . iy ( i ) .«< io> > . (x (1>.w<20>>.
1(U (II ,*(2<j) I IFUMT.NE.Ol GO TO 100 NX-9 NY-10 NIJ- 1 NR-3 RETURN
100 CONTINUE 9ETA 1 ■ , 37 Wl«1000.
RET A2-S7 , 6 W2*1S00.
RET A3*. 608 W3»565.
LINKG1-1.3T2
LINKG2-2.86*
kmo.163.
CSTABA.l. /7.12ft
CPR0*l./57.3
CVPIN«16.
BDM001 - 1 ,2C MOROOl-lOO' .
R0R002-1.17
HOR002-200.
CG0K-.296
C
c differential equations c
V ( 1 l-KMR-Y ( 1 |
V (?) -X (3)
V ( 7| -M3-Y ( ? I V (6| -M2-Y ( 7 |
V ( 5 1 -MOM DO 1*7 16 1 V (6) -Y (5 1 V ( 7) -WDM0D1 *Y (6)
V(ft)»Y(7)
V ( 9 ) -M 1 • Y ( 7 )
c
C SUMMING POINT FQUATIONS
c
V<10)-LINKM»X(2>-CSTABA-XU I V ( 1 1 1 -Rr T A 7-X (6) -X ( 3l
V < 1 ?> -RET A’-Y ( 8) -X (6 )
V ( 13I--X (5) »CVPIN-RDM0D1»X(?I V ( 16) — X (6) »X (5)
V ( lS)a- X I 7 1 .RDMQD2-X ( 1 )
V(16)— XIB).X(T)
V ( 1 7) -X ( 9) -CGOK-V ( S)
Y(la)»PETAl*Y<10l-X(9l V ( 1 9) *U ( 1 ) -Y ( 7)
60 C OUTPUT CQU *tT IONS
C
V <?0) *LINMr,?*CRPD*X ( 1 )
V (21 ) »L iNKr.Z-CRPD-XDOT ( 1 )
V ( ?2 ) -KMO-|_ INKG2-CRP0* <L INKOl-XDOT ( 2 1 -C5TABA*XD0T ( 1 ) )
65 RETURN
END
Figure 111. Program Listing for Actuator Equations
197
M.-I aha,
ifc'aafifiriV
>
Figure 112. Controller Block Diagram
NAME l: t for controller
XCw)=ROLL OFF FILTER STATE (STRUCTURAL FILTER)
XC 2)=NOTCH FILTER STATE 1 (STRUCTURAL FILTER) XC(3)=NOTCH FILTER STATE 2 (STRUCTURAL FILTER) XC(4)=STATE OF COMPENSATOR 2 XC(5)=STATE OF COMPENSATOR 1 XC(6)=COMMAND INPUT SHAPING FILTER STATE XC ( 7 ) =NORMA L ACCELERATION FEEDBACK LAG FILTER STATE RC(1 )=ELEVATOR COMMAND OUTPUT FROM CONTROLLER UC(l)=PILOT INPUT (CENTER STICK)
UC(2)=NORMAL ACCELERATION FEEDBACK INPUT UCO^PITCJf, RATE FEEDBACK INPUT
Differential Equations
yn = % • ycd)
x (2) = x (3)
C X
ic(3) = u,d2 • yc(2)
V4) * "a ■ yc(3)
*c(5) ■ ®cl ' yc(4) xc(6) = ouj • yc(6)
ic(7) = 0,3 ■ yc<7)
Summing Point Equations
yc<l> * xc(2) - XC<1) + (2Cn/oin) • xc(3> + (o>c2/l»n2> • yc(2) yc(2) = xc(4) - (2Cd/«ld) • xc<3> - xc(2) + (u>c2/«>c4> • yc<3) yc(3) - xc(5) - xc<4> + (i»c1/«ic3) • yc(4) ycM) ■ KvKp. y c<5) - xc<5)
yc(5) ■ Kq . «c(3) + K„z • xc<7> - 2 Kcs • *c<6> yc(6) * uc(l) - xc(6) yc(7) - uc(2) - xc(7)
Figure 114, Controller Equations
200
Response Equations
r (1) » x (1) c c
Values of the Parameters
UUj = 6 |
0) „ , =1 Cl |
Cd * .6 |
H II CQ O |
© 00 II CM O 3 |
ti)f = 120 |
X X) ii • CD CO |
“c3 = 4 |
|
U)3 = 4 |
i»c4 * 27 |
|
Knz = *561 |
u) = 86 n |
|
= 19.97 |
cn . .05 |
|
KF = . 25 |
<«d = 84 |
SUBROUTINE SIMkC
CDC 6600 FTN V3.0-P355 OPT-1 11/13/73 00.67.37
SUBROUTINE SIHKC C SIHKC 6600 VERSION
COMMON V(6l>*M(70)*NXtNY*NRtNUf INlT«IFLA0*MODE»r<6l»70)«T»irC OIMENSION XOOT (7)*X(7)*Y(7(tU<3)
S REAL KCS»KQ*KNZ»KV*KF
EQUIVALENCE <X0OT(n*M(l))«(Y(n»W(0>)»Um*«(lS)>((U(l)»IM22)> IF (INIT.NE.O) 60 TO 100 CALL 6AINTAB< IFC»KF>
NX»7
10 NR*1
NU>3 NY«7 RETURN
100 CONTINUE 15 Wl«6.
kcs"1 •
KQa.83
*3«4.
KNZ-.561
20 KV-19.97
KF *. 25 *C1«1.
HC2-80.
WC3-*.
25 WC4-27.
WN-86.
XIN-.05
WO«84.
XIO-.6
30 WF« 120.
C XOOT EQUATIONS v(n»wF*y<n
V <2)«X 13)
V(3)«MD«WO»Y(2)
35 VU)«WC2*V<3)
V<5)»MC1»Y<4)
V (6 ) »M1*Y ( 6 )
V(7)«M3*Y(7)
C Y EQUATIONS
*0 V(8)«X(2)-X(1)*( ( 2.*X IN) /WN) *X ( 3) ♦ ( ( VD#NQ)/ (NN*NN) )»Y<2)
V ( 9 ) *X (4 ) - ( ( 2 »*X ID) /WO) *X ( 3) "X (2) ♦ (WC2/WC4)*Y (3)
V(10)*X(5)_X(4) ♦ (WC1/WC3) *Y (4)
V < 1 1 ) »KV*KF*Y (5) “X <5)
V(12)»K0*U(3)»KNZ*K(7J-2.»KCS*XI6>
65 V(13)=U(1)-X(6)
V(14)»U(2)-X(7)
C RESPONSE EQUATIONS V ( 15>»X ( 1 )
RETURN
50 END
l
Figure 115. Program Listing for Controller Equations
Figure 116. Plant Block Diagram
NAME LIST FOR PLANT
XP( 1)=XS(1)=PITCH RATE GYRO STATE 1 XP( 2)=XS(2 'PITCH RATE GYRO STATE 2 XP( 3)=XS(3)=NORMAL ACCELEROMETER STATE XP( 4)=XV(1)=ANGLE OF ATTACK (ALPHA, RAD)
XP( 5)=XV(2)=PITCH RATE (Q, RAD/SEC)
XP( 6)=XV(3)=STABILATOR BENDING (ETA1)
XP( 7)=XV(4)=STABILATOR BENDING RATE (ETA 1 DOT)
XP( 8)=XV(5)=FIRST VERTICAL BENDING (ETA2)
XP( 9)=XV(6)=FIRST VERTICAL BENDING RATE (ETA2DOT) XP(10)=XV(7)=STABILATOR ROTATION (ETA3)
XP(1 l)=XV(8)=STABILATOR ROTATION RATE (ETA3DOT) X'°(12)*XV(9)=GUST ANGLE OF ATTACK (ALPHA, RAD) XP(13)-XA(U=POWER ACTUATOR STATE (DELTAE, RAD) XP(14)=XA(2)=SECONDARY ACTUATOR INTEGRATOR STATE XP( 1 5) =XA (3) "SECONDARY ACTUATOR (SERVO VALVE) STATE XP(16)=XA(4)=SERVO AMPLIFIER STATE
XP(17)=XA(5)=DEMODULATOR (FILTER) STATE (SECONDARY ACTUATOR)
XP(18)=XA(6)=WASHOUT FILTER STATE (SECONDARY ACTUATOR) XP(19)=XA (7) "DEMODULATOR (FILTER) STATE (STABILATOR ACTUATOR)
XP(20)=XA(8)= WASHOUT FILTER STATE (STBBILATOR ACTUATOR)
XP(21)=XA(9)= ACTUATOR INPUT FILTER STATE
RP( 1)=RS(1)=PITCH RATE GYRO OUTPUT
RP( 2)=RS(2)=NORMAL ACCELEROMETER OUTPUT
UP( l)=UA(l)=ELEVATOR CONTROL INPUT TO ACTUATOR
UP( 2)=UV(4)=WHITE NOISE INPUT TO GUST FILTER
Differential Equations x
s
A x + B u
S S S3
x = A x + B u
v v v v v
x = A x + B u
a a a a a
Output Equations
r = C x + D u s s s s s
r = C x + D u v v v v v
r
a
C x + D u a a a a
Figure 117. Plant Equations
204
Interconnection Equations «.<!> ■ rv(l)
us(2) = rv(2)
uv»> ■ ra(l)
uv(2) = ra(2)
V3> * ra<3>
Plant Outputs
rp(l) = rg(l)
V2) ' rs(2>
Figure 117. Plant Equations (Concluded)
subroutine simkp
COC 6600 FTN V3.0-P355 OPT-l 11/13/73 00. 4?, 37
5
10
15
20
25
30
35
40
45
50
55
SUBROUTINE SI*KP C SIMKP 6600 VERSION C
C F-4 PLANT (SENSOR-VEHICLE-ACTUATOR)
C
COMMON V (41 ) *4(70) ,NX,NV*NR«NU, INIT. IFLAS.MOOC.F (41 ,70) ,T» IFC DIMENSION XSDOT (3) ,XS(3) tXVDOT (91 .XV (9) .XADOT (9) ,XA<9> »RS<2? • 1 US(2) ,RV (2) ,UV (4) ,RA( 3) »UA( 1 ) » U(2)
DIMENSION AS(3.3) ♦BS(3*2) *CS<2.3) .OS (2.2)
1 . AV (9*9) .BV(9.4).CV(2«9).DV(2»4)
2 ,AA(9,9>,SA<9,1),CA(3.9).DA<3.1)
COMMON/DTAPE/HARK (20). LOCATE. INSERT .null DIMENSION I SEN (20) • I ACT (2ft) • IVEH(20 )
EQUIVALENCE (XSDOT(l).H(l) ).(XVD0T(li»V(4)).(XAD0T(l).tMl3)).
1 <RS(1)*W(22)),(RV(1),W(24)),(RA(1),W(26))»
2 (US < 1 ) ,V<29) ).(UV(1)»V(31)).(UA(1)»M(3S) ).
3 (XS(1).V(36)).(XV(1).V(39))«(XA(1).M(4S)).
C INITIALIZE C
IF( INIT.NE.O) GO TO 100 NU ■ 2 NR * 2 C
c read inputs PROM sensor, VEHICLE, and actuator c
READ (5*299) ISEN READ (5,299) IVEM READ (5,299) IACT 299 FORMAT (20A4)
CALL TAPE(L0CATE.ISEN.7>
WRITE (9,299) ISEN
READ (7) T.nSX.NSR.NSU, ( ( AS ( I , J) , I>1 ,NSX) .Jal.NSX) ,
1 ( ( B S ( I,J) »I»1»NSX) ,J»1 . NSU) ,
2< (CS(I.J) ,I«1.NSR) ,J«1.NSX) *
3 < <DS ( I » J) . I»1 »NSR) , J»1 »NSU) •
CALL TAPE(L0CATE,IACT,7)
WRITE (9,299) IACT
READ (7) T.NAX.NAR.NAU, ( (AA( I.J) . I«1 ,NAX) , J-l.NAX) ,
1 ( (BA(I,J) ,I«1.NAX),J«1,NAU),
2((CA(I,J) ,1-l.NAR) *J«1.NAX) ,
3( (DA (I ,J) , I«1,NAR) , J«1,NAU)
CALL TAPE (LOCATE, IVEK, 7)
WRITE (9* 299) IVEH
READ (7) T.'lVX.NVR.NVU, ( ( AV ( I , J) , I»1 »NVX> .J-l.NVX) .
1 ( (BV(I,J) , 1*1 ,NVX) , J»1 , NVU) ,
?( (CV(I,J) ,I>1,NVR) ,J«1,NVX),
3 ( <DV( I , J) . I»1.NVR).J«1,NVU)
NX = NSX ♦ NVX ♦ NAX
NY = NSR ♦ NSU ♦ NVR ♦ NVU ♦ NAR ♦ NAU C PRINT out matrix QUADRUPLES FOR SENSOR. VEHICLE, AND ACTUATOR C
IF< IFLAG.NE.O) GO TO 102 WRITE (9,112)
Figure 118. Program Listing for Plant Equations
206
s
[>
*
I
s
L
;
i
t
112
103
60
65
70
102
111
106 75
100
C
C COMPUTE DIFFERENTIAL EQUATIONS
C
80 C SENSOR DYNAMICS
DO 200 1*1, NSX
V<I)«0.0
DO 201 J*l,NSU
201 V < I > «V < I > ♦QS < I » J) *US ( J)
85 00 200 J*1»NSX
200 V(I)>V(I)*AS(IiJ)*XS(J)
C VEHICLE DYNAMICS 00 2021*1 »NVX 11*1 *NSX
90 V(II)*0.0
DO 203 J«1,NVU
203 V<m*V<II>*BV<ItJ)*UV(J)
DO 202 Jsl.NVX
202 V<II)*V(in*AVCl.J>*XV(J)
95 C ACTUATOR DYNAMICS
DO 20* 1=1. NAX II*I*NSX*Nv/X V(II)*0.0 DO 205 Jal.NAU
100 205 V<II)*V(II)*BAC#J)*UA(J)
DO 206 Ja 1 .NAX
206 V < 1 1 » * V < 1 1 ) ♦ AA(I.J) • XA ( J)
C
C COMPUTE OUTPUT EQUATIONS
105 C
C SENSOR OUTPUTS
DO 26 1*1 #nSR II = I ♦ NX V ( 1 1 ) = O.ft
110 DO 27 J*).'ISX
FORMAT (21HC0NTINU0US QUADRUPLES) CONTINUE
CALL MPRSUS»NSX»NSX»NSX.NSXtT»6HAS CALL HPRS (RStNSX t NSU*NSX*NSUtTt6HBS CALL MPRS(CS'NSR»NSX*NSRtNSX»T'6HCS CALL MPRS<DS»NSR*NSU.NSR»NSU*T.6HDS CALL MPRS <AV»NVX»NVX»NVX*NVX#T »6HAV CALL MPRS(RV.NVX»NVU*NVX*NVU*T.6HBV CALL MPRS (CV*NVR»NVX »NVR*NVX»T»6HCV CALL MPRS <DV*NVR«NVU»NVR*NVU*T *6HDV CALL MPRS (AA*NAX» NAX tNAX»NAX»T»6MAA CALL MPRS (RA.NAX»NAU.NAXtNAU»T.6MBA CALL MPRS<CAtNAR»NAX»NARtNAX*T»6HCA CALL MPRS(0A»NAR»NAU»NAR.NAU*Ti6HDA GO TO 106 MRITE(9«lll>
FORMAT (18HDI6ITAL QUADRUPLES)
GO TO 103 CONTINUE RETURN CONTINUE
Figure 118. Program Listing for Plant Equations (Continued)
♦
I
207
SUBROUTINE SIMkP
115
120
125
no
135
140
145
27 V < 1 1 ) « V(II> ♦ CS(I»J> • XS(J> 00 26 J«1 »NSU
26 van > van ♦ Dsa.j) * usu>
C VEHICLE OUTPUTS
00 26 1*1 »NVR II ■ I ♦ NX ♦ NSR
van ■ o.o
00 29 J*ltNVX
29 van * van ♦ cva.j> • xvu>
00 28 J*1»NVU
26 van * van * ova.j> • uvui
C ACTUATOR OUTPUTS 00 30 I-l.NAR II * I ♦ NX ♦ NSR ♦ NVR
van«o.o
00 31 J*1»nAX
3i van * van ♦ caci.j) • xa<j>
00 30 J-ltNAU
30 van * van ♦ oa(i.j> • uaij> c
C INTERCONNECTION EQUATIONS
c
c SENSOR INPUTS
V (29) * RV ( 1 )
V (30) * RV (2)
C VEHICLE INPUTS
V (31 ) ■ RA(l)
V (32) * RA (2)
V(33) ■ RAO)
V (34) * 11(2)
C ACTUATOR INPUT V(35> ■ U(l>
C
C PLANT OUTPUTS C
V (36) * RS(1 )
V(3T> * RS(2)
RETURN
END
Figure 118. Program Listing for Plant Equations (Concluded)
208
igure 119. Overall System Block Diagram
/
NAME LIST FOR OVERALL SYSTEM
X( 1)=XP( 1)=XS(1)*PITCH RATE GYRO STATE 1 X( 2)=XP( 2)=XS(2)=PITCH RATE GYRO STATE 2 X< 3)=XP( 3)=XS(3)=NORMAL ACCELEROMETER STATE X( 4)=XP( 4)=XV(1)=ANGLE OF ATTACK (ALPHA, RAD)
X( 5)=XP( 5)=XV(2)=PITCH RATE (Q, RAD, SEC)
X( 6)=XP( 6)=XV(3)-STABILATOR BENDING (ETA1)
K( 7)=XP( 7)=XV(4)=STABILATOR BENDING RATE (ETA1DOT)
X( 8)=XP( 8)=XV(5)=FIRST VERTICAL BENDING (ETA2)
X( 9)=XP( 9)=XV(6)=FIRST VERTICAL BENDING RATE (ETA2DOT) X(10)=XP(10)=XV(7)=STABILATOR ROTATION (ETA3) X(ll)=XP(ll)=XV(8)=STABILATOR ROTATION RATE (ETA3DOT) X(12)=XP(12)=XV(9)=GUST ANGLE OF ATTACK (ALPHA. RAD) X(13)=XP(13)=XA(l)=POWER ACTUATOR STATE (DELTAE, RAD) X(14)=XP(14)=XA(2)=SECONDARY ACTUATOR INTEGRATOR STATE X(15)=XP(15)=XA(3)=SECONDARY ACTUATOR (SERVO VALVE) STATE X(16)=XP(16)*XA(4)*SERVO AMPLIFIER STATE X(17)=XP(17)=XA(5)=DEMODULATOR (FILTER) STATE (SECONDARY ACTUATOR)
X(18)=XP(18)=XA(6)=WASHOUT FILTER STATE (SECONDARY ACTUATOR)
X( 1 9) =XP(19)=XA (7 > -DEMODULATOR (FILTER) STATE (STABILATOR ACTUATOR)
X(20)=XP(20)=XA(8)=WASHOUT FILTER STATE (STABILATOR ACTUATOR)
X(21 )=XP(21)=XA(9)=ACTUATOR INPUT FILTER STATE X(22)= XC(l)=ROLL OFF FILTER STATE (STRUCTURAL FILTER)
X(23)= XC(2)=NOTCH FILTER STATE 1 (STRUCTURAL FILTER)
X(24)= XC(3)=NOTCH FILTER STATE 2 (STRUCTURAL FILTER)
X(25)= XC(4)=STATE OF COMPENSATOR 2
X(26)= XC(5)=STATE OF COMPENSATOR 1
X<27)= XC(6)=COMMAND INPUT SHAPING FILTER STATE
X(28)= XC(7)=NORMAL ACCELERATION FEEDBACK LAG
FILTER STATE
R( 1)=RP( 1)=RS(1)=PITCH RATE GYRO OUTPUT R( 2)=RP( 2)=RS(2)=NORMAL ACCELEROMETER OUTPUT R( 3)= RC(l)=OPEN LOOP TEST OUTPUT FROM CONTROLLER
(LOOP BREAK POINT)
U( 1)= UC(l)=PILOT INPUT (CENTER STICK)
U( 2)= OPEN LOOP TEST INPUT TO ACTUATOR (LOOP
BREAK POINT)
U( 3)=UP( 2)=UV(4)=WHITE NOISE INPUT TO GUST FILTER
Differential Equations
x_ = A x + B u P P P P P
x„ = A x + B u c c c c c
Figure 120. Overall System Equations
210
J
Output Equations
r_, = C x + D u P P P P P
r = C x + D u c c c c c
Interconnection Equations
u (1) = u(3) r |
|
u (2) = r (1) P c |
if MODE = 0 |
up(2) = u(2) |
if MODE = 1 |
up{2> = u(2) |
if MODE = 1 |
uc(l) = u(l) |
|
uc(2) = rp(2) |
|
uc(3) = rp(l) |
|
Overall System Outputs |
|
r(l) = rp(l) |
|
r(2) = rp(2) |
|
r(3) = rc(l) |
if MODE = 1 |
Figure 120. Overall System Equations (Concluded)
I '
211
./r.kit'idiMi;
.
►
t
«l|T«*».*W.-FY' j ■!■■¥» J'WWTWH - VI, . ■" . '■•11' 'M l ■ . 1 1
\
SUBROUTINE SINK COC 6600 FTN V3.0-P355 OPT»l 11/13/73 00.67.37.
5
10
IS
20
25
30
35
40
45
50
Sc
subroutine sink
COMMON V (41 ) ,W<70) .NX.Nr»NR,NU»INIT,IFLAG.MODE»F<41.70> .T.IFC DIMENSION XCOOT ( 7 ) , XPQOT (21 ) ,XC ( 7 ) . XP ( 21 ) ,RC ( 1 > ,RP (1 > *UC (3) .UP (2) ] ,U<3>
DIMENSION AP <21 ,21 ) . BP (21, 2), CP (2.21). OP (2.2)
1 ,AC(7,7),3C(7,3)«CC(1,7),DC(1«3)
DIMENSION I HEAD (20 )
COMMON/DTA^E/MaRK ( 20 ) .LOCATE, INSERT .NULL 0IMENSI0N I CON (20) , I PL (20)
EQUIVALENCE (XpqOT (1 ) • W < 1 ) ) , (XCOOT ( 1 ) ,W (22) ) • <RP<1 ) ,W<29) ) ,
1 (RC ( 1 ) , W < 31 ) ) < (UP ( 1 ) ,W ( 32) ) , (UC ( 1 ) *W ( 34) ) ,
2 (XP(1 ) ,W(37> ) » (XC(1> »H(5S> ) , (U(ll *«(6S) 1 IFdNIT.NE.O) GO TO 100
C
C INITIALIZE
c
NU ■ 3 NR ■ 3
C READ LABELS TO FIND PROPS'* QUADRUPLES READ (5, 107) ICON 107 FORMAT < 20 A4 )
READ(5,107) IPL C READ PLANT quadruples
CALL TAPE(L0CATE.IPL.7>
WRITE <9. 106) (IPL (I). I -1.20)
106 FOPMAT(1X,?OA4)
READ (7) T.NPX.NPR.NPU. ( ( AP ( I , J) » I«1 .NPX > . J»1 .NPX) ,
1 < (BP(I.J) ,I«1.NPX) ,J«1,NPU),
?((CP(I.J).I«1.NPR),J»1,NPX),
3( (DP ( I, J) • 1*1 *NPR) ♦ J«1 ,NPU)
C read CONTROLLER QUADRUPLES
CALL TAPE (LOCATE* ICON, 7)
WRITE (9, 106) (IC0N(I)»lal«20)
READ < 7 ) T ,NCX ,NCR t NCU, (<AC(I»J)»I»1»NCX)»J«1»nCX)»
1 (<BC(I,J),I«1.NCX),J«1,NCU),
2 ( (CC ( I • J) , I»1 .NCR) , J«1 ,NCX) ,
3((DC(I,J).I»1*NCR) »J»1,NCU)
IF(IFLAG.NE.O) GO TO 102 WRITE(9.112)
112 FORMAT (1X21HC0NTINU0US QUADRUPLES)
103 CONTINUE
CALL MPRS(AC,NCX,NCX,NCX,NCX*T,4HAC CALL HPRs(9C,NCX»NCU,NCX,NCU*T,4HBC )
CALL MPRS(CC,NCR,NCX,NCR,NCX*T,4HCC )
CALL MPRS (DC,NCR*NCU,NCR,NCU,T , 4HDC )
CALL MPRS(AP,NPX,NPX»NPX,NPX,T,4MAP )
CALL MPRS(BP,NPX.NPU.NPX,NPU,T,4HBP ) call MPRS(CP*NPR,NPX.NPR,NPX,T,4HCP >
CALL MPRS(DP,NPR,NPU,NPR,NPU,T,4HDP )
GO TO 104 102 WRITE (9,111)
ill format (Ixibhdigital quadruples)
GO TO 103 104 CONTINUE
NX * NCX ♦ NPX
Figure 121. Program Listing for Overall System Equations
'i
212
f.
{
SUBROOT INE SINK
NT ■ NCR ♦ NCU ♦ NPR - NPU RETURN
100 CONTINUE C
C COMPUTE DIFFERENTIAL EQUATIONS
C
C PLANT DYNAMICS
DO 203 I-UNPX II - I v < 1 1 ) a 0.0 DO 202 jal.NPU
202 V(II) a VCII> ♦ BP ( I * J) a UPtJ) DO 203 Jal.NPX
203 V < 1 1) a veil) ♦ AP(I,J) * XP(J) C CONTROL 0YNAMICS
DO 201 Ial.NCX II ■ I ♦ NPX V<1I) a o.O DO 200 Ja] .NCU
200 vein a vein ♦ bc<i»j> • ucej>
DO 201 Jal.NCX
201 vein a vein ♦ Acei.Ji * xciJ)
V
c compute output equations c
c plant outputs
do 213 1*1 .NPR
II ■ I ♦ NX
vein a o.o OO 212 jal.NPx
212 vein a vein ♦ cpji.j) *xp uj
00 213 Jal.NPU
213 vein ■ vem ♦ dp<i»j)»up<j> c control outputs
00 211 1*1 »NCR
II * I ♦ NX ♦ NPR
vein a o.o
00 210 J * l.NCX
210 “ V<II> * ccei»j)»xcej>
DO 211 Ja 1 .NCU
c 2,1 v<ii) * vein ♦ Dcei.j)*ucej)
c interconnection equations c
c flant inputs
v<32) a seen
IFeMOOE.EQ.n ve32) a u (2)
VC33I ■ ues>
c control inputs
V(34) a U ( 1 > ve35l a RP ( 2 )
V(36) a PP(1)
c
C F*»4 SYSTEM OUTPUTS
c ________
V f 37) a PP(U
V <3S> a PP(2| ve39> a Reel)
RETURN
end
Figure 111. Program Listing for OveraU System Equations (Concluded)
213
Stability and Frequency Response Performance
Parametric analysis by software was carried out to relate the poles and the frequency response of the F-4 longitudinal control system to the sample time of the controller.
The following parameter set was used:
Sample time: 0, 1/1000, 1/160, 1/80, 1/40, 1/20 sec Coefficient word length: 24 bits
Figures 122 and 123 show the sample time root locus in image s-plane and in the z-plane. Only closed loop vehicle poles (rigid body and three bending modes) are illustrated. The computer output of the closed loop poles ape given in Figures 124 through 129.
Table 12 shows the phase margin as a function of sample time at the first gain crossing (rigid body margin). Tables 13, 14 and 15 show the gain margins at 1st, 2nd and 3rd phase crossings respectively. The computer output of the margins for sample times T = 0, 1/160, 1/80, and 1/40 sec are presented in Figures 130 through 133 respectively.
Figure 134 shows the frequency table of the F-4 open loop control system for T = 0, 1/160, 1/80 and 1/40 sec. The bode plots of the F-4 open loop system are given in Figures 135 through 142. In Figures 142a and 142b the bode plots of the F-4 open loop system are overlayed.
\ ■ 1 ' ■. ' " 1 7 T
Frequency ratios were computed and are displayed in Table 16 . The frequency
ratio is defined as follows:
p a ^
(406)
where
uunq = Nyquist frequency = ^ = rad /sec T = Sample time
w . = Frequency of interest in the frequency response (rad/ sec)
In terms of frequencies. Equation (406) can be written as:
f = 2 p f . s 1
(407)
where
f = Sampling frequency (samples/second) s
f. = Frequency of interest in the frequency response (cycle /second)
For finite bandwidth systems (fictitious) of bandwidth the shape of the frequency response is retained if p = 1.
In the following we define a criteria for retaining the shape of the frequency response of continuous systems when it is discretized.
Frequency response shape invariance criteria is said to be satisfied if the deterioration in phase margin at rigid body crossover frequency does not exceed a prescribed value (3 degrees assumed here) and the attenuation, at the bending frequencies is not less than a prescribed value (6 db assumed here).
For actual systems, p = 1 is not sufficient to satisfy the frequency response shape invariance criteria for the frequency of interest uh. Table 16 and Fig- ures 135 through 142 show thac p = 2 is satisfactory for this purpose. Using this value and Equation (407), the sampling rate for the F-4 system becomes fs = 100 samples /second, for the longitudinal channel. In terms of the sample time root locus in the z-plane (Figure 123) this means that the frequency response shape invariance criteria is satisfied if the significant modes of the system (z = x + iy) lies in the first quadrant of the z-plane. That is
U)iT«F-/2
In this case, for tu^ = 160 rad/sec
T **72) (160) n -Q1 sec* (408)
This result will be referred to as the "1st quadrant rule".
The following conclusions can be drawn from this parametric study:
/
• Sapnple time root locus in the image s-plane (Figure. 122) shows that the damping of the rigid body and 1st bending modes are increased for sample times 0 <T s 1/80 sec. The second and third bending mode dampings are decreased for sample times 0 <T * 1/80 sec.
• The phase margin as a function of sample time for the first gain crossing frequency (rigid body) shows that (seejTable 12) a loss of 4 degrees occur for T = 1/80 sec. sample time. Note that the sub- system approach predicted the same loss for this crossing (see Figure 92 of previous section).
• Second bending mode gain margin is reduced as sample time is i increased. It becomes less than 6 db at T = 1/80 sec (see Table 14).
To maintain a minimum of 6 db gain margin the sample time shoulcTbb less than or equal to T = 1/100 sec.
215
A
i n' ■nii AH iiiiriii’ifaiflfel --
Figure 122. Sample Time Root Locus in the Image s -Plane (Closed Loop F-4)
216
2 <1/801
F-4 Vehicle Modes)
POLES OF T HF SYSTEM
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216
124. Closed Loop Overall System Poles (T = 0 sec)
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.996728666 *00 ,9967?066F *00 .9951 3*64f *00 .9951 376*6*00
.259594606*00 . >59594406*00
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-,45761976:*00 -.457619766*00 -.734511496*00 -.73451 1*96*00 -.53*970486-01' -.53*970*86-01
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Figure 125. Closed Loop Overall System Poles (T = 1/1000 sec)
219
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Figure 126. Closed Loop Overall System Poles (T
1/160 sec)
220
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- • 35371 751E *00 |
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Figure 127. Closed Loop Overall System Poles (T * 1/80 sec)
221
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0.
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. 1 ?56*371r .0 3 .1256*3715.03 .1756*371*. 03 0.
-. 1 1 7*745*r «03 • 1 1 7*745*F«03 - . I 4*4?*56f .0?
.14*42*585.02
-.64***4*75.02
.*4**64*75.02
-.40*48*115.02
.40*48*135.02
0.
-.7*6077445 .01 .7**077445.01 0.
0.
-.8075531*5.02
.8075531*5.02
-.455424**5.02
.455424**5.0?
0.
0.
008® TNG
-.818571 72£ .00 -.651B2***E»00 -.58742130E*00
-. 33*635235 .00 -. 33*635?3t *00 -.733*50>,7E.OO -.733*54*7£.00 -. 24286832F «00 -. 24286* 32C.00 -.86435616F-01 -.*6435*16E-0i
-.541 58R77E *00 -.54158477E.00
-.31 02* 1 37C-01 - • 3102* I 37F-0 1 -.153I8718E-01 -.153I87ISF-01
FREO
.21877??IE»03 .16570?*0E*03 • 1 553*81 5E«01
. 12*6*6*6F *03 . 1 ?*66606E *03 . 24265685E *02 . 24265685E «02 .72631637E.02 •72631637E.0? •406*23*4E.92 .40e*2364E«02
■42531851E.01
.42531851E«01
, 80 74*?05E *02 . *074*205E»02 . 4S60*?0?E.02 •45604202E.02
Figure 128. Closed Loop Overall System Poles (T a 1/40 sec)
222
•*811*1
- - ... . 88l06iir
— ■ V. , Jj-ML
P01.E5 ftF Tm.- STST7M
•uii * ,'Kin>-w
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D»“OiNG
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0.
-.11 70*. 7*8F- i * -,|| TO- Ti.«7- | * J5-S57F-05
,95*7a?99F-fl* .95*7*5997-0* .1601 9665F-0 1 , 1001O665F-" | -.S669o*69F-n| ,|077o*76F.fl0 .797S*61BF*nn .2975*6|8E*00 -.*68|*3*3F .00 -. *681*3* 1E*00 - .6**9**90E *00 -.1677a7*3E*00 -. 1677o7*1E*00 • 66 I 7 17 38f *00 .66171/18F.00 .7101 30**f *00 .«181*182F*60 . 8 167 T6S6E *00 , 1 069*6*<>E *00 .1069«6*OE*00 -,S9|*1761E*00 -.59|8|T61F*00 ,951259*2E*00 .95111209F.00
0.
0.
0.
0.
0.
0.
- . »9*l on 7 fr -07 . 700 1 OO 7 7r -07
0.
0.
-.’57211*57.00 • 557711*5* .00
-.1*61661 1 r • 00
. 1«0 1 0.0. 3 |1 .00
0.
-.6**68**67.00 , S0<.6«<.a0r >00 -.796755657.00 . 700 755*5* *00 0.
0.
0.
-.«9|?l**7r.00
,*9121**77.00
- . 7736027*iF *00 .771607767 *00
0.
0.
.0600771. or. 00 5*977**7 .00
•7S6S0*1SF*00
.TS6S0i.3Gr.00
-.77M0’OOr.00
-.77007997.00
-.731 17OOS7.00 -.731 3700SF *00 .912*6S70F*00 .O17i.6G70r.00
. 1 lOlRTior.oo . i ioi*7|or.oo -.6310*»9*F*00 - . M30**98r .00
. 1 0***280F-0 l . 1 0***7*OF-0 1
.393330*37.00 . 303330*1F *00 .60309J69E*00 .60309J69F *00
. 7037 779SF *70
• 703777OOF *00
• 72521891E r 00 ,72521891E*00
.89761318E *00 .89761338E *00 . *3**6860E *00 ,o3*86fl60E*00
5-»t.*NE
Or BE
0.
-,T79ST*|6E*01 -,779S7*16E*03 - . 17570042F *01 -. 18S1 1723F *03 -.1991 17?3E*03 -.9I?2615?E*0? -,917?*3STE*07 -.S719OB09F *0? -,**6*’*57E*02 -. 19667|9*F*0? -. 1866710*F*0? -.1011 365*E *07 -.101 H6S*E*0? -.8771n637E*01 -. 707SO899E .01 - • 707SO899E *01 - .6*?563*SE *01 - .6*?S61*SE *01 -.60*5*1 71E-01 -.*013*1 39E*0 1 - . 15630619E *01 - , 7 1 59*7 1 1 E *0 1 -.2159071 IE-01 - ■ 1 3*69960E *0 1 -, 1 3*699605 *01 -. 10000000E*01 -,9978*160E*00
I«4»S
0.
.67*318*37.0? ,67831 9S 3r *0? .67*31**37 .0? 0.
0.
-.S7*6*709r.nl ,57*8*7097 .0 1 .678318*37 .0? 0.
-. l*?56O00r .0?
.1*5560007 *0? -.*91915177.0? .*919151 77.0? «6?931 8*3r *o?
-.3608*61|r.o?
• 3608*61 1 r . 0 ? -.**1051697-01 .**1051697.01 0.
0.
0.
- . 7902650 37 *0?
.29026503**0?
-.*51756187*0?
.*51256187*0?
0.
0.
o*“8tNr.
-.99676775F *00 - ■ 99676T75F *00 -.981 90568F *00
-.99802O56F *00 - . 99807056F *00 -.67**7)817.00
-. 79*6*7fl8E *00 - • 79*6*788E *00 -.?01 3B5?7F*00 -.701 18527E *00 -.139755737*00 -.1911 195*F *00 -.19111 95*F *00 -.60 61 *5*3F *00 -,606|8s*8F »00
-.7*?05) 60E-01 -.7fc?05 I 60E-0 1 -. 2983639*E-0 1 -.?993639*E-01
FBEO
• 7821 02) I E *0 3 .7821021 IE *03 .131 79299E*03
,9|*07293E*02 .91*07293f*02 ,851033*8E *02
, 73*6*7*7E *02 ,?3*8*7*7E *02 .50220*26E*02 ,50220*28E*02 .63**1 101E*02 ,36762279F*02
• 36762279E *02 .106001 13E*02
• 1 0600 1 1 3E *02
. 791 06750F *02 • 291 06750E*02 •*51*5737E*02 .*51*5737E*02
Figure 12S. Closed Loop Overall System Poles (T = 1/20 sec)
223
Table' 12. |
Phase Margin vs |
. Sample Time (First Crossing) |
|
Sample Time T (sec) |
First (lain Crossover (t'(.) (rad /sec) |
Phase Margin (deg) |
1 .oss of Phase Margin (deg) |
0 |
9. 97 |
74. 70 |
0 |
1/160 |
9. 97 |
72. 62 |
2. 08 |
1 / 80 |
9.97 |
70. 87 |
3. 83 |
1 / 40 |
9. 96 |
67. 38 |
7. 32 |
Table 13. |
Gain Margin vs. |
Sample Time (First Crossing) |
|
Sample Time T (sec) |
First Phase Cross- over Frequency Gain Margin (igj ) (rad /sec) (db) |
Loss of Gain Margin (db) |
|
0 |
35. 465 |
14. 04 |
0 |
1 / 1 G 0 |
33. 87 |
13.49 |
0. 55 |
1/80 |
3 2. 35 |
12. 99 |
1.05 |
1/40 |
28. 20 |
11.60 |
2. 44 |
Table 14. |
Gain Margin vs. |
Sample Time (Second Crossing) |
|
Sample Time T (sec) |
Second Phase- Crossover (X02 ) (rad /sec) |
Gain Margin (db) |
Loss of Gain Margin (db) |
0 |
96. 91 |
11.17 |
0 |
1 / 1 60 |
97. 21 |
9.43 |
1.74 |
1/80 |
97. 61 |
5. 96* |
5. 21 |
1 /40 |
80. 38 |
6. 56 |
5. 61 |
93. 36 |
5. 29 |
5. 88 |
Less than 6 d b
224
* - - ■.■■■•— •.r—^-^jg^m,
Tablt* lb. Gain Margin vs. Sample Time (Third Crossing)
tuple Time !’ (see) |
Third Phase Crossover (t0.^) (rad /see) |
Gain Margin <db) |
Loss of Gain Margin (db) |
0 |
105. 2 |
3 2. 73 |
0 |
1 / 160 |
1 78. 2 |
2.4. 48 |
7. 2 4 |
1 /HO |
1 (j 1 . GG |
i 0. 12 |
1 3 . G 1 |
1 /4 0 |
1 52 . 0 |
3. H |
2H. 03 |
171.0 |
1 2. 3 |
20. 43 |
|
2 24. 0 |
11.1 |
21. G3 |
Less than G d I)
225
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226
E13S6310
FREQUENCY RESPONSE
(OUTPUT 3/INPUT
GAIN CROSS |
OVfR = |
9.9691 |
RAO/SEC . |
PHASE* 70. |
870 |
||
0. |
phaSF |
CROSS |
ovfr = |
32.353 |
RAO/SEC |
« |
GAIN* * i2.0no |
180.00 |
PhaS'E |
CPOSS |
ovfr = |
BO. 575 |
RAO/SEC |
• |
GAIN* -21.511 |
0. |
PHASE |
CPOSS |
OVE» = |
07.614 |
RAn/SEC |
« |
GAIN* -5.9630 \ |
ibo.oo |
PHASE |
CPOSS |
OVFRs |
131.30 |
RAO/SEC |
• |
GAIN* -20.525 |
0. |
PHASE |
CROSS |
OVER = |
161.65 |
RAO/SEC |
« |
GAIN* -19.121 |
180.00 |
phase |
CROSS |
OVFRr |
214.50 |
RAO/SEC |
GAIN* -48.839 |
|
ino.oo |
dhaSE |
CROSS |
OVFRr |
2S«. 10 |
RAO/SEC |
« |
GAIN* -75.405 |
iR0.no |
phaSF |
CROSS |
OVFRr |
2BR. 1R |
RAO/SEC |
« |
GAIN* -48.828 |
0. |
PHASE |
CROSS |
OVrR = |
339.64 |
RAO/SEC |
• |
GAIN* -21.271 |
ino.oo |
PHASE |
CROSS |
OVrR = |
371.55 |
RAO/SEC |
t |
GAIN* -20.104 |
o. |
PHASE |
CROSS |
ovfr= |
409. OB |
RAO/SEC |
• |
GAIN* -16.411 |
ino.oo |
PHASE |
CROSS |
OVfR = |
42B.22 |
RAn/SEC |
• |
GAIN* -24.446 |
o. |
phasf |
CROSS |
OVFRr |
466.92 |
RAO/SEC |
* |
GAIN* -14.248 |
GA I N CROSS |
0'/E° = |
491.79 |
RAO/SEC . |
PHASE = -68. |
853 |
||
GAIN CROSS |
OVCR = |
SU .20 |
RAD/SEC • |
PHASE* -43. |
328 |
||
0. |
PHASE |
CROSS |
OVrR = |
509.45 |
RAO/SEC |
GAIN* 1.2203 |
|
0. |
PHASE |
CROSS |
OVFRr |
538.19 |
RAO/SEC |
GAIN* -14.154 |
|
0. |
PHASC |
CROSS |
OVFRr |
633.19 |
RAn/SEC |
GAIN* -20.102 |
|
0. |
phaSF |
CROSS |
OVFRr |
6P0.67 |
RAO/SEC |
GAIN* -20.03H |
|
ino.on |
PhASF |
CROSS |
OVFR* |
717.52 |
RAn/SEC |
GAIN* -48,867 |
|
inO.OO |
PHASE |
CROSS |
ovfr* |
730.85 |
RAn/SEC |
GAIN* -55.77R |
|
1 BO . 00 |
PHASE |
CROSS |
OVFRr |
790. 7B |
RAO/SEC |
GAIN* -40.623 |
|
0. |
PHASF |
CROSS |
OVFRr |
868.15 |
RAO/SEC |
GAIN* -21.130 |
|
0. |
omacf |
moss |
OVFRr |
931 .65 |
RAn/SEC |
GAIN* -15.451 |
Figure 132. F-4 Control System Open Loop Frequency Response (T -- 1/80 sec)
F14S6410 T = l/<*0 0. L. NO MOLn ( RE3UENCY RESPONSE
(OUTPUT 3/INPUT 2)
04 I M |
pposs |
0 tfCR = |
9.OS40 |
RAO/SEC . |
phase - 47.374 |
|
0. |
PHASF |
CROSS |
0'/rP = |
2«.ms |
RAO/SEC « 0 A I N = -11.SR4 |
|
n. |
PM4SF |
CROSS |
OVFR- |
no. inn |
RAO/SEC . GAINs -4.S771 |
|
0. |
phase |
CROSS |
OVFRr |
93.356 |
RAO/SEC . GAIN = -5,2913 |
|
180 |
.00 |
ShASF |
CROSS |
OVFR- |
104.63 |
RAO/SEC * C,A!N= -12.927 |
ino |
.00 |
PHASE |
CROSS |
OVFRr |
122.59 |
RAO/SEC . GA IN= -42.144 |
1«0 |
.00 |
phasf |
CPOSS |
OVFR- |
14S.R0 |
RAO/SEC . GA 1 N= -13.404 |
o. |
dmasf |
CROSS |
OVFRtr |
is;>.qr |
RAO/SFC . G A I N = -3. 8397 |
|
dhASF |
CROSS |
OVFRr |
171.13 |
RAO/SEC . GA I N= -12.334 |
||
0. |
phase |
CROSS |
OVER r |
224.12 |
RAO/SEC . GAINe -11.130 |
|
gain |
CROSS |
OVER = |
’SO. AS |
RAD/SEC . |
°HASEe -42.33P |
|
gain |
CROSS |
OvER = |
RAO/SFC . |
PHASE* 44.441 |
||
n. |
d*-aSe |
CROSS |
OVER-. |
2Sh , 0 s |
RAO/SFC . GA IN* 11. m3 |
|
n. |
dmase |
CROSS |
OVl Or |
282.38 |
RAO/SEC • GAINe -12.448 |
|
o. |
dwaSE |
CROSS |
OVFW- |
360.21 |
R An , SEC . GAIN= -18c 340 |
|
mo |
.m |
PHASE |
CPOSS |
ovfr- |
348.08 |
PAn/SEC • GAINe -26.104 |
0. |
ouasE |
CROSS |
OVERr |
428.54 |
RAO/SEC . GAIN= -26.969 |
|
0. |
phase |
CROSS |
OVER- |
465.54 |
RAn/SEC . GAINe -15.341 |
|
04 IN |
CROSS |
OV'EPz |
<*R i . n<» |
RAO/SEC . |
RHASEe -64,814 |
|
04 IN |
CPOSS |
OVFP = |
sn ms |
PAD/SEC • |
OHASEe -46.874 |
|
o. |
PM ASF |
CROSS |
OVER t |
509.1 8 |
RAn/SFC . GAINe 1 .378S |
|
0. |
phase |
CROSS |
OVFR- |
879.84 |
RAO/SFC • GAIN* -IS. 7 |
|
0. |
Phase |
CPOSS |
OVFRr |
48h.ni |
RAO/SFC • GAIN* -28. OSS |
|
GAIN |
CROSS |
OVF» = |
754.1 4 |
o AO/SEC • |
“HASEe 114.69 |
|
04 1S| |
CROSS |
OVFR = |
7f>n.i7 |
R*n/sEr . |
°HASEe 114.08 |
|
n. |
phase |
CROSS |
0VF« - |
7 49 .38 |
RAO/SFC t GAIN= -4.3443 |
|
1 on |
.on |
phasf |
CROSS |
OVFR t |
8S 1.63 |
RAn/SEC . GAIN* -m.80h |
n. |
PH4SC |
CROSS |
OVFRr |
nno.74 |
RAO/SEC • GAIN= -22.16S |
|
0. |
PHASF |
CROSS |
OVFR- |
940.98 |
RAO/SEC ♦ GAINs -16,47) |
Figure 133. F-4 Control System Open Loop Frequency Response (T = 1/40 sec)
S»MPL£ T!«e FPFECT ros r-<. r>Pr* lOO3 Output < i>/
ppwwpiwpppnwp
iPlPWT^
— K©r-«/af\*a©A.a«a © © a pn c inr« <co < o 4 — 4 ©©or* 4— — 4r*© — ff* ^cci^i\'- t r*
o — 4 r* c r* r* c / o r a r • ••••• •••••©©aifaaAc.
• • • • • • • • • • * • •
4 if if if © ©• © e* f- t* © cr a a o c. ©oce©©cc ~ ~ . r\ a a aiAAAAA— > — ©ar-f** r*r“cr*
c <y o o o o o o 0 c* o 0 0 o~— - — — — — a a 10 r* f* »* 4
a — « a r^ if — jK<rr^ff rir a a «A©Aa©A©©aA 4 4 r* ^ ©<ye©e<rc©©©a**ar*iraaA*
a 4«—C(r~— c— aa r if © r* 0 Aif©4a*©A-— a-cta 4 k c ir - a — c — Aj o 4 v <r 4 4 r*1 if a a a
• • O © — a- 4 © © Aj cr 4 c © A <r 4— A4©*r'©A if A © r-> 0 cva-— r>r^.-.kf0A.if©,ifr-a 4 <r a a r-
cc • ••••• • • • •(? x m a if ©
O ff ff « I cr f*- r* © © 4 if if 4 4 4r'r'r-f\J(\rv.— • • • • • — — — a AA.f"'fr~r’>r'A rv — — • •
ififooco© irif if c c c if ire c cccci/iroc cifccocircifccifirifoircifififc © c c © 4 a if © © A © 4 — © © 4 rv ©0c?0»o — r-ircA© — cr"©©**©**©— ifaiftv— ca.4©4a— a©© c c — Aj a n a* 4ifif©r*©00e — A4ir©A©© — r'4©aa — nirr*©A4Acr‘©aAif0f'^— © •
• <v.rtjf\ir\jr\.(\jfvjrvjr'f^r'r'r'f^r<4 44 4ifirifif©1©'©©A-A r* a a cr 0 —
©a»aA©4'"i‘* — ©«*' — r* c-* o a 4 if 4if©(?A©— f*-©Aj©a©oAj0©4-T — 4**— ajc 4 A. c^ c a- ©
f ©■ © — ir cr a f cif c if • • • • • • • • • • • • • • • • f" 4 © © — *4
• • • • • ©iffx^4'4rvja^rrvJ • • • • • • •
©©©©©©AAaaa a © c ©e©ccoc©e — — — — — — aaa A A Ai A AAA — — C c 4 C if •> a r* © 0C*00O0aaa0Ca~ — — — — — — — — — — — — — — — — — — —— — — — — — — — — — — a ■* 'T I ^ ^ ^
#V — k j f ic 4M?fi-c4i(t j—.j'r'fA©© 4aa<\jcrarv—»4 ff'4iriff--4 « j® — 00 — — o — aa 4if©A0^ir©4cA 4 a - 1 5 r < .4orjn*n4c<
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Preceding page blank 249
Table 16. Frequency Ratios
Gust Response Ratio Performance
The overall closed loop F-4 longitudinal control system model is utilized to develop the state and output variance to a gust input. Variances are com- puted with the continuous controller first, and then with digital controllers at sample times of T ■ 1/1000, 1/160, 1/80, 1/40 and 1/20 second with the "continuous gust input". The gust was represented by a filtered white noise.
A first order filter with bandwidth of 46 rad /sec was used.
The namelist of the overall system states, outputs and inputs are given in Figure 143 to facilitate the reading of Table 17. The variance ratios for the vehicle states as functions of sample times are plotted in Figure 144, and tabulated in Table 17. Figures 145 through 149 show the computer outputs for sample times of T - 1/1000, 1 /160, 1/80, 1/40 and 1/20 sen. respectively. Variances corresponding to the 2 8 states and the 3 outputs given in Figure 143 are printed out. Also variance ratios are computed in per unit and in db.
The following conclusions can be drawn from the variance response ratio * performance evaluation:
• For the vehicle modes sufficiently away from the half sampling frequency, the variance response ratio shows monotonicaUy increasing behavior for increased sample times. (See aaZ and a 2 responses for all T *1/160, in Figure 144.)
• For the vehicle modes close to, or greater than, one-half the sampling frequency, the sampled variance response ratio shows an increasing envelope and alternating behaviour. For these cases, the sampled response may not be sufficient to see the peaks of variance, and the intersample variance response should be evaluated.
• Sample time T = 1 / 1000 sec is used to verify the continuous and discrete models. For this small sample time, continuous and discrete models produce practically the same variance. Note that the variances of digital controller states do not correspond to the variances of continuous states, because under the Tustin transform the continuous system state and digital system state are not the same (see Section III). However, the output of the controllers must agree (response r(3) for T= 0 sec and T = 1/1000 sec), and they do.
To keep the bending-modes displacement and displacement rate rms values within a 125 percent envelope, a sample time of T = 1/100 sec. is needed.
For a short period mode, the T = 1/20 sec. sample time is sufficient.
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Figure 146. |
Variance Response Ratios for Sample Time T = |
1/160 sec |
257
C#sT lNu8u* System y*»!*Nct
•3Jo»«E 00 •417*48 -Ol •(HUE Of
•1**988-03 •437TK-01 •IUM 0* •103388-0* •844481 01 ***3»»C-04
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Figure 147.
rtM VARIANCE •i*»eo8-oi |
RESPSNIf RATIO T» >128008 *01 |
RESPONSE RATIO (Ob T • • 12300E-01 |
•ill J7E 00 |
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Variance Response Ratios for Sample Time T * 1/80 sec
258
•MMMMMtWWMUklMiaiia
C»NTlNwluf IftTlh VAIMNCt |
OllCItTE lYlTIM VARIANCE' t* aaooot»w |
ftlfMNIt RATH ,8*10001*01 .. |
|
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|
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Figure 148. Variance Response Ratios for Sample Time T =
259
. . « s i
n|H)|[ qq
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1/40 sec
as
CONTINgOuS iTEf- variance |
DISCRETE system variance T. •BOOOOE'Ol |
RESPONSE SATIS Ti «80000E*01 |
RESPONSE RATIO (Do T. • 30000C*01 |
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Figure 149. |
Variance Response Ratios |
for Sample Time T = |
1/20 sec |
260
Effects of Computational Time Delay on Longitudinal Control System Stability
A reduced model was developed for the parametric study of computational time delay effects. In this model the actuator and gust dynamics are modified (a third order actuator and a second order gust filter). The same model is used in the simulation tests. The overall closed loop parametric system quadruples were developed for stability analysis. The modeling details are fully presented in Appendix C. The following set of computational delays was used for parametric study:
Td = 0. T /4, T/2, T sec.
The sample time was fixed at T = 1/40 sec.
The effect of computational delay is studied by computing the poles and zeros of the overall closed loop system. Figures 150 and 151 show the time delay root locus of overall closed loop system in the z-plane and in the image s-plane respectively. Figures 152 through 155 show the system quadruples corresponding to Tj = 0, T/4, T/2 and T sec, respectively. Figure 156 shows the system closed loop quadruple with T^ = 0 and obtained using SIMK subroutine to verify the results of HSIMK subroutine which models the time delay into the system (see Appendix C). Figures 157 through 160 show the poles of the model for these delay times.
The following conclusions can be drawn:
• The first bending mode stability is greatly affected by a delay. At one sample delay the real part of the 1st bending mode is reduced from two to one in the image s-plane.
• The rigid body and the second bending mode dampings are increased up to the half-sample time delay. The trend is changed, however, for sample times greater than half-sample time delay.
• Computational time delays should be less than one -fourth the sample time to maintain adequate bending mode stability.
261
RIGID BODY
Figure 151. Time Delay Root Locus in the Image s-Plane for Sample Time T = 1/40 sec (Overall Closed Loop)
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Figure 155. Closed Loop Overall System Quadruple Using HSIMK for Sample Time T = 1 / 40 sec. , and Time Delay Td = T
270
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Figure 155. Closed Loop Overall System Quadruple Using HSIMK for Sample Time T = 1/40 sec. , and Time Delay T^ = T (Continued)
271
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3-COLU*N |
4-COLUMN |
5-COLUMN |
6- COLUMN |
7-COLUMN |
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0. |
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-.13376017*00 |
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2-o o* |
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-.?55144?C-04 |
0. |
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5-BO* |
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6-90* |
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7-90* |
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Figure 156. Closed Loop Overall System Quadruple Using SIMK for Sample Time T = 1/40 sec., and Time Delay 1^=0
273
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6. |
0. |
0. |
A. |
14-tOW |
.4J74345F-61 |
.44316547 *66 |
•07*464*7-6* |
I. |
0. |
0. |
A# |
17-tOW |
*. |
0. |
• . 7000666I *60 |
.4|4476Tf *06 |
-.4*434*07-6* |
-.756*6637 *46 |
•11364557*01 |
It-tOW |
0. |
0. |
6. |
,344?374f*66 |
.74344447-6? |
-.64361137*60 |
.06*50407*60 |
14-tOW |
o. |
6. |
6. |
-.5*4*0477*0? |
-.405*0457 *60 |
-•514004 77*0? |
.770063*7*6* |
?o-tow |
i. |
6. |
1. |
6. |
0. |
0. |
|
*l-tOW |
o. |
0. |
0. |
A. |
0. |
0. |
.07530067*06 |
**-tQW |
o. |
0. |
0. |
0. |
0. |
6. |
6. |
*3-tOW |
0. |
0. |
0. |
A. |
0. |
6. |
0. |
?»-COlU*«»J ?3-rOLU«N
l-tow |
t. |
0. |
*-tow |
0. |
A • |
3-tow |
0. |
0. |
4-tOW |
0. |
0. |
4-tOW |
0. |
0. |
6-tOW |
O. |
0. |
7-tQW |
0. |
0. |
o-tow |
». |
0. |
o-tow |
0. |
|
16-tOW |
0. |
6. |
It -tow |
0. |
0. |
If -tow |
*. |
0. |
13-tOW |
1. |
0. |
14 -tow |
0. |
0. |
15-tOW |
0. |
0. |
16-tOW |
«. |
0. |
1 7-tOW |
•10555577*01 |
|
it-tml |
-.jiwnc-.i |
.54450137*60 |
i o-tow |
-.mijur.oi |
.71507317*6* |
*o-tow |
.64155377*00 |
|
fl-tow |
•6556**67-01 |
|
f**tow |
..0 |
0. |
73-tow |
e. |
•46476)47*00 |
Figure 156. Closed Loop Overall System Quadruple Using SIMK for Sample Time T = 1/40 sec., and Time Delay T^ » 0 (Continued)
274
<?• .?seooF-oi>
l-POM |
.4568?Pn-0l |
-*13l 3367E.00 |
,*04l0*5F-0> |
?-pom |
. |494958f-0? |
-•l 084788C-0 | |
- • 3 3???09f -0 3 |
3-*0M |
,1841 146C *00 |
-•III 7789C *01 |
-.38813447-01 |
4-POM |
,47900 1 ?f -05 |
-.3I1153IC-0* |
-.8548678E-05 |
5-POM |
,468?4fc9f -03 |
-.3394I7QC-0? |
-.I090?87f -07 |
4 -POM |
,?4?5340r-01 |
- • l *04 1 50f * 00 |
-. 3719094C-07 |
7-POM |
.753585 3C .01 |
-•1 839745C »0? |
- . 79*904 3F -0 1 |
8-POM |
.7874908E-0? |
-.7048894C-01 |
-.117??84r-04 |
P-POM |
-.71?8380C-0l |
.5170174C.OO |
-.5748805F-07 |
10 -POM |
,53?68*7f-03 |
- . 3B63830C-0? |
-.1719997F-0* |
1I-P0W |
-,«?40343F»00 |
.4716494C*01 |
-.5648473F-01 |
1?-P0M |
0, |
ft. |
. |970077f-0-» |
13-POM |
0. |
0. |
.8359074F-04 |
14-POM |
-.74l?78#*C-01 |
.1749904' *00 |
0. |
15-POM |
-,8l8?553f-0? |
.5934778C-01 |
0. |
10-POM |
-,9?7l584f-03 |
,*7?4644E-0? |
0. |
1T-P0M |
-.3188008C*00 |
0. |
ft. |
18-POM |
-. 398985 00 |
0. |
0. |
1 9-POM |
-,l?9044?f.0? |
0. |
ft. |
?0-POM |
- , 75468 1 Of *00 |
0. |
ft. |
?l -POM |
- , 3?98898f -01 |
0. |
ft. |
?l-*0M |
.l?97999f *00 |
0. |
ft. |
?3-P0M |
0. |
0. |
ft. |
M*TP|« H |
i T ■ .7S000F-0I » |
||
1 -COLUMN |
7-COLUMN |
3-C(KU«N |
|
1-POM |
0, |
. 1 OOOOOOf • 0 1 |
0. |
l-POM |
, 1 OOOOOOF ♦ 0 1 |
0. |
0. |
3-POM |
.7439447F-01 |
, *70 1 ?4>E *00 |
0. |
•-tolumn |
9-COLUMN |
10-COLUMN |
|
l-POM |
0. |
0. |
0. |
?-»0M |
0. |
0. |
0. |
3 -POM |
0. |
0. |
0. |
|*-COLUMN |
14-COUIMN |
1 7-COLUMN |
|
l-POM |
0, |
0. |
ft. |
?-POM |
0, |
ft. |
0. |
3-POM |
0. |
0. |
. 1 0 000 00r . 0 1 |
?7-colu*n |
73-C0L'8#N |
||
l-POM |
0. |
0. |
|
?-P0M |
0, |
0. |
|
3-POM |
0. |
0. |
|
UTP]K l |
• T • ,?8000f-0ll |
||
1 -COLUMN |
? -column |
3-column |
|
l-POM |
0. |
0. |
0. |
?-P0M |
0. |
0. |
0. |
3-POM |
-.I378746f*00 |
0. |
0. |
Figure 156. Closed Loop Overall System Quadruple Using SIMK for Sample Time T = 1/40 sec. , and Time Delay Trf = 0 (Concluded)
275
I
PQUES OF THF system «MM ■ .9813815982
Pr»L I Mir, DAMPING FREQ
.10537890E-07 0.
-.66?i9?*?e-ei o.
-.I023|703E«00 0.
-.299RS895E.00 -.8RS44982'--01
-,29985R95E»00 .88544962F-01
-.72349506E-01 -.585421S8F.t)0
-.7?3*OS08E-01 .58542)5RF*00
.5*97 1 427E *00 -.209790'.05«00
.5497142TE.00 .20979040f»00
-.51098258E.OO -.642903R1F.00
-.S109*,?S8E*00 .8*2903*15.00
.R8048SI2E.00 0.
.84908S98E«00 - . 18R9720) F .00
,8490*S98E«00 . 18R972olr *80
,9047a 1 90E*00 0.
.91 R8470*E»00 0.
-,4| 17a781E«00 -.851957405.00
-,4| I7A76IE.00 .853957*05.00
-.71 5h* 1 34F -00 -.838227*85 .00
-.71 SO 4 1 34 f *00 .83R22748F.0O
.9753I327F.00 0.
.981341f.OE.00 0.
.9«1341<SOE.OO 0.
-.9748541 1F» 00 .30T59162E»00
-.9748541 1E»00 ,30759162E»00
- . 1 28921 92E »00 .S7001159E.00
-.12892192E.00 .57003159E.OO
• 93427529E ♦ 00 .588385 75E»00
.93427829E.00 . 588385 75E«00
-.622I9872E.O0 .82122352E.00
-.S2219472E.00 ,82I2?352E»00
.98076777E.O0 .86573600E«00
.98076777E.OO .86577600F *00
- .43*331 84F.00 .948048425 . 00 -.434331 84E.00 .94804842F.OO -.748319915*00 .95BB93S4E.00 -.74631O91E.00 .95889354E.00
.
S-9l ANr
Pr*L |M*r,
-.274F.971F.07 0.
-. 10R5 J778F.03 ,l?5<S437lr.03
-.91 187| 855 *02 . |25887T|F .03
-.47159291F.02 -.118874*75.03
-.471S3291F.02 . 1 18874*7r.03
-.224R.540E »02 -,47922440r *02
-.224R.540F.02 ,879774805.07
-.2l2l490IE«02 -.14587010^*0?
-.2121-90 IF *02 .I45H7OI0 -.02
-.787R79H3F.01 - , *949 74 77' « 1?
-.7878 .98 7F • 0 1 .89h974.Tr. 02
-.8011 .881 F ♦ 0 1 0.
- . 57874 1 07C .0 1 -.7R57571?r.ni
-. 5787410 7F* 01 .78575712. *01
-.400177835.01 0.
-.33941 721F.01 o.
-.21 774BR0F • 0 1 - . RORO 7707r ,02
-.2| 774RR0F.01 .R0807707r.0?
-.18791.790F.0I -.98532487r,02
-.1879I.090F.OI ,98532482< *02
-.9998. 240F.00 0.
- . 75757 1 4 3F ♦ 00 0.
- • 7575 7 1 4 7F . 00 0.
DAMPING FRFO
-.853847505.00 .18808*235 *0 3 -.58730O38F.00 , I 5528257F .03 -.774741 48F *00 • I25R4487F .0 3 -.7747414RE.00 .I2584487F.03 -. 3I473803F *00 , 7 1 548885F .02 -.71473403F.00 .71548685F.02 -.82408244E .00 . 7574788 7F. 02 -.87408744F.00 , 7574 788 IF .02 -,87499»?4F-0l ,900190191.02 -.8749 J824F-01 ,900 790 1 91 .0?
-,5918877?r.00 .97487880F *01 -.59188 * 7?" .00 .97487880F.01
-.284007785-01 .808318815.02 -.284007285-01 .808318815.02 -. 17 3905 72F -0 1 .98547083F .02 -.17 3905725-0 1 .98547087F.02
004*8401
000*840)
apoamfn r pot f 5 APPAR5NT POL F 5
pGl£5 OF THF SYSTEM
8MAX .
*981361 5982
2-pL. ANE
iMAfi
0.
0.
0.
0.
-.?*??6388f«00
•?*226388F*00
-.5719?070r.00
•S719?0?0r*00
-.1798*?|0r.00
• 1 7984?) 0r*00 -•628575l0r»00
•67857510F • 00 -• I 778*751f»00
• 1 778*75 IF *00
0.
0.
0.
-*6?3366«qr*00 •67336689r*00 - .85 7689 1 6r «00 .857659|6r.00 0.
0.
0.
OiWf N6
-.80327Q81E»00 -•8032798IE»00 1 l?09*09f»00 *•11 ?09*09E'00 . 956895 32E-00 . 956895 32E«00 -.631969*6E‘00 **63196O46E»00 •978?178*E*00 *97«?1784f *00
* • 7*58 1 566C ♦ 00 -.7*551S66E*00 -•***03«??e*00 *.***03»2?e»00
5-»L*NE
Or AL
.151 35055F -0? ••**97077 9E -6? -.80896Q67E-01 1096607*E»00 -. 3?67778BE*00 * • 376777B8E-00 -. 6*515* 75E-0I - .6< 51 5*7SE-0 1 .597SP923F.00 •59?5?923E«00 *.51?57?ft9E*00 *.51 ?S7?69E »00
• 8 1R0Q991E *00 .R7R0O997F.00 ,R60*6S1?F.OO
• 90*7* | 90E *00 .»16?:>*67E.OO
*.697?o|*3(r.on - • 697?q 1 *3E *00 *.*?75SS7*f.oo -.*?05557*F*00 .97531 9RIE *00 .981 3*1*OE*00 .9R136160F.00
Or AL
- . 75977107E *03 -.7161 73I0F.01 -. 1005a7|6E«03 *.88*|4T3SF.O?
-.35981 T94F.O? *.3598| 79*F*07
-.72097339F.0?
*.??097339f .o? -. 191 7| 757F.07 -.191 1 1 75?F *07 - • 87750* ] 6F * 0 1 - .8 7759* 1 6F * 0 ) -. 618 70 01 5F .01 -.61R7801SF.0I
* . 60 J 1 7861F -01 *0017787^.01 -. 1*997*86f.01 - . 76 75*8 1 *E .0 I -. 767508 ifcf-, o)
- . 1 573* 7 1 OF *0 1 -. |983*710F.0I
- • 99959* 0 IF * 00
-. 75757K3E-00
-.757571*^.00
IMAO
0.
. 1 756637|r*03 • 1 ?566371r*03 .175663711 .03 -. 1001*3] ?r *03 . 1<10I*3I?F»0.1 - • 67775060F *02 •67775060r.0? -. 1 1 7871 7*c 0?
. 1 1 7871 7*r»o? -.901 95577r .07 . 90 1 95677r . 0*’ - . 8 76*05 70* ♦ 0 1 .816*0570f«01 0.
0.
0.
-,96*8*10lr.p?
■ 96*8* ] 8 1 F . 0? -.81 71S876F «07 . 8 1 7358 76r ♦()? 0.
0.
0.
DAMBjMG
-.86*539035.00 -.6?*S9775F»00 - . 575*?689F*00
- .338 1 3931 F«00 -.3371 3931 E»00 -.31 185073E*00 -.71 185073E.00 -.851B7799F .00
- • 851 87799J.J0 -•92*66*20F-01 -.9?*66*?0E-01 -. 59**950 3E* 00 - . 59**950 3E *00
-.777I9087r.il
- . 7771 9087F-0 I -.7** 1 o J 50E-0 1 - • ?** I 0 1 50E-01
FRED
. *0675*75E»00 .*0675*75E*00 j5755*753E»00 •5755*753E*00 .6192?053E*00 •61922053E»00 .81 I07193E*00 .81 107I93E-00 .85676206E«00 . 85676206E*00
•93530085E-00 •93530085E-00 . 95 l6?0?0E »00 •9S167020F»00
FREQ
. ?500**35E *03 . 160960675*03
• 15365068E-03
• 1 06*1 1 15E-01
• 106*1 1 15E-03 .70858705E-02 .70858705E-0? .22505*1 IE»02 .22505*1 1E*02 .90583605E«02 .90583605E-02 .10*01 772E-0?
• 10*01 77?E*02
•96521269E-02 .96521 269E*02 .81=>60089E*02 . 8 1 260089E ♦ 02
Figure 158.
Loop System for sam»ie ™
= 1/ 40 sec
277
POLES or Tnr system
J will'-. 4 ) .; i,n J
i mjillHIlLlin mil
HlWrH— 6M' ■
UMAX > .9813615982
2 -°l*ne
R*AL
• 255670 1 0E-02 -• 19358787E-01 -.8853O140E-01 - . 1 2 154051 E *00 -.36527078E*00 -.36527078E*00 -.40287580E-01 -•40287588E-01 . 6316Q353E *00 . 6316435 3C*00 -.50027944E-00 - .500? 1944E *00 .82734530E*00 . 82734538E *00 . 860445 1 ?E *60 . 904761 90E *90 . 91 378348E *00 -,69296239E»00 • . 69296?39E*00 -.43?7S564E*00 -.43276564E.00 •9753OR83E*00 • 481 34 1 60E *00 •981 36160E*00
IMAG
0.
0.
0.
0.
-.35445648**00
• 75445648* *00 - .57739068* *00
•57739068* *00 -.13224476**00
• I 3224476* *00 -.6I663569*»00
,61663569**00 • • 1 95094137 *00 . 195094) 3**00 0.
0.
0.
-.59699547* *00 .59694597* *00 -.85806356* *00 .95806356**00 0.
0.
<).
OAMPJNG
-.7|761»33E*00 -.71 761 233E *00 - .695991 T Of -0 1 -.69599IS0F-0I .978781 35E *00 .978781 35E*00 -.63000>?0f *00 - . 63000220F»00 . 97330568E *00 .47330468E*00
- • 75761*30F*00 -.75761830F.OO -,45031085*»00 -,45031085E*00
FREO
,50895277E*00 .50895277E*00 .578794l4E*00 .578794)4E*00 .64518776E*00 .64538778E *00 . 79402649E *00 .7940?644E*00 . 85003645E *00 .95003645E*00
• 91465899E *00 ,91465899E*00 .9610 1 536E*00 ,961 0 1536E *00
5-°LANF
R*AL
Imag
DAMPING
FRE3
-.23876932E |
♦ 03 |
0. |
|
-. 157764 35E |
•03 |
.17566371* |
• 03 |
-.96974489E |
♦ 02 |
.12566371* |
•03 |
-.84787143E |
♦ 02 |
.17566371* |
• 03 |
-.27016002E |
♦ 02 |
-.94846567* |
• 0? |
-.270I6082E |
♦ 02 |
.94846567* |
• 0? |
-.21877337E |
♦02 |
-.65619072* |
• 07 |
-.21677337E |
♦ 02 |
.65618072* |
• 07 |
-.17514157E |
♦ 02 |
-.92547667* |
• 01 |
-. 1 75161 57E |
♦ 02 |
,9?547657r |
• 01 |
-. 92255 381E |
♦ 01 |
-.90094095* |
• 0? |
-.92255381E |
♦ 01 |
.90094095* |
.07 |
-.64990421E |
♦ 01 |
-.92630779* |
>01 |
-.6499042IE |
♦ 01 |
.92630779* |
>01 |
-.6011 788 IE |
♦ 01 |
0. |
|
-. 4003738 3E |
♦ 01 |
0. |
|
-.36064650E |
♦01 |
0. |
|
- . 35681 590E |
• 01 |
-.97. 19107*. |
02 |
-.35601590E |
♦ 01 |
.97719107*. |
0? |
-. I5905956E |
• 01 |
-.91516391*. |
02 |
- . 15905956E |
•01 |
.91516391* |
0? |
— , 1 0000444F |
• 01 |
0. |
|
-.75257143F |
• 00 |
0. |
|
-.TW57143E |
• 00 |
0. |
-.782?3035E*00 - .6 1 0945?9E *00 -.55703755**00 -.77394 778* * 00 - . 27394278E *00 -. 31 6??3 16E*00 -. 31 6??7 I 6E*00 -.9045B»43E*00 -.90458743F *00 -. 1 01 86675F *00 -.10186675**00 -,67474501**00 -,57434501E*00
-. 36677922F-0 1 -,766779??E-0I -. 1 95089 73*-0 1 -. 1 9508873F -0 1
.7017108„E*03 .1587318 E*07
• 151 31 327F *03 .98619145F *0? .98619146E *07 .69167409F *07 .691674097*07 . 1 9367804F *02 . 1 936 7804F *07 .90566706F *07 .90565206F *07 . 1 13I5577E *07
• 1 131557?F*02
,97?83565E*0? .97?83565F*0? ,81571 908F *07 • * 1 631 908F*0?
Figure 159. Poles of the Overall Closed Loop System for Sample Time T *1/40 sec and Time Delay T^ = T/2
278
POLrS or Tmc sySTTm
.^MAX r ,'HHhl 4 9HP
v\°>'
\vf-
7-3i *Nf
«(.
IM45
i'/i«oiNr.
FHf3
. I^Sl^MOF-OP i '.q^mf-01 -. in?ot iO*F*nn -.?44P ih4?f*0() |
o. O. n . |
||
-# V*>* >?H0F *00 |
- . 7 1 « I ^r,s<- *on |
-,8598<'a?0t .00 |
,89881 887E *00 |
*»?^4.n?«0F*O0 .*?« \ o*<.r ♦no |
.37181?05r.00 0. |
-.859fl?o?0F *00 |
.8988 l887f.00 |
,S04 ntulr.n? |
-.'ll lOOMO?, .on |
,b?3 3-,<, OTf-O? |
.81 3079) OF .00 |
,Sft4 7144 If -OP . 74S1044SF *01 |
.4-1 00081?- .00 o. |
,B?3l88975-0? |
.81 307910E»00 |
-,4ftnQ> QM4F *ft0 |
- ,88?‘iflBq8-' ♦ 0 0 |
- • 5991 7 >9?f .00 |
. 80783988E *00 |
-.4MQ1 QH4F *no . hf,04*.c | ;>f • on |
,88740«0<,l .00 n. |
- . 509] 7’9?r .00 |
• 80783988F .00 |
i4lQ4"A7?r *00 |
- . Sfc?n«. 1 t fcr .on |
.O4-0857H5F.00 |
* 87380 790E .00 |
??f *no ,Q04 t ioof *nn .oo^7*i''.aQr ♦no |
. 78 ’Ob | | 8i .00 n. 0. |
,0ftQB578Sr ,00 |
,87380799f .00 |
- , Tnmoigf *00 |
-.5718507Sr.OO |
-.78988757F *00 |
. 03 1 53807E *00 |
- ,7iss »Qc,nF*nn .QTsii i??F*no |
.sm^’Sr.oo 0. |
- . 78988 757r *00 |
.O31S3897F.00 |
-.4l*S7f»A if ♦no |
.oo |
-,885l88p7r.00 |
• 98031 1 ?Of .00 |
-.414^ J*61F*00 .9*1 i^!*>nr *no .9*1 V-1AO^OO |
.877710851 *00 0. 0. 5- |
-,88S1888tE.OO JL»Nr |
• 080 3 1 1 ?Op *00 |
Pr al *1U? >S 10f • O’l |
I M 4 5 0. |
04MB T |
FOE0 |
-. | 070 77gOP *01 |
. | »8881t|i .03 |
-.551 1695?F *00 |
. 18581 705F.03 |
-.91 1fl4| 74F *n? |
. | ?504-1 7 | r .01 |
-.587811 14, f. 00 |
. 15533750E *03 |
-.5?9?^4 7«r *n;> |
.17588171- .01 |
-.10818898c .00 |
,136358??F.03 |
- . ph 14 ->h44F ♦ r p |
-.0)855857' .0? |
-.793SP587F -00 |
, 05878 708F » C? |
-.PA14PH44F *n? * • P 1 91 SSS7F *0P |
,Q| 855057‘ «0? 0 . |
- . 7935?58?F .00 |
.05878708F .07 |
-• 1 9*7 WJSf *0? |
-,8’50751 1' .0? |
-. 798BS'.70F *00 |
.85805757E .0? |
-. 19S7171SF *0? -. 1 0 70 '->00 IF *0? |
.875Q751 It -0? 0. |
-.798858 FOT .00 |
, 85805 75 71 .0? |
-.H79V1QSPF *01 |
-.BBS 10550- .0? |
-,OM88M01r-0| |
, 88988, ’51f .0? |
- , 0 7Q1 >OSPF ♦ n I -.*•011 >AMlf .01 |
,885 1055 9' .0? 0. |
-.08848 ,01( -01 |
.8898875 IF . 0? |
- .S404O4 1 ?F *0 1 |
-.11 ??q?u8i. .o? |
-,83170|88F.00 |
. 17887358F.07 |
-.S40414 l pr .0 1 - . 40 01 llMlr .01 -.191 l07<,*r .n 1 |
. 1 1 ??q?J4-r .0? 0. o . |
-.83170188F.OO |
, 1 7887358F ♦ 07 |
- . ?RlAn*4?F .0 1 |
- . 007808095 . 0? |
-.785730885-01 |
• 0978 I 8 38F .07 |
- , ph if, jf,4?f *n i -,999fl^4l4f.00 |
.007808091 .0? 0. |
-.7M573988F-01 |
.997H1838F *0? |
-. 79S4'.449>: .00 |
- , 8 1 70<,7pq, .(j? |
-.97H3801 Tr-o? |
.8 1 798 1 70F .07 |
-.79*4'449F*00 |
,8 | 758779- .0? |
-.0781801 7r-o? |
,81 ?98| 70F .07 |
-.7SPS7|41F*00 -.7SPS7i4ir.no |
0. 0. |
Figure 160.
P0}^ of Overall Closed Loop System for Sample Time T and Time Delay Trf = T
1/40 sec
279
SECTION VI
CONCLUSIONS AND RECOMMENDATIONS
The objective cf this study were threefold: 1) development of theoretical analyses and mathematical models for digital flight control systems, 2) development and documentation of computer analysis programs, and 3) demon- stration of their use by a detailed parametric study for computational require- ments. The major emphasis has been on analysis and software development.
These objectives were primarily met. The analyses and model developments as well as testing and demonstration of their use including the parametric study are documented in this report. The developed programs have been docu- mented in AFFBL-TR-73-119, Volume II. The parametric study was con- ducted on a representative tactical fighter-bomber aircraft, the F-4.
In the following, the results and recommendations for future studies pertaining to the work in the area of analysis and synthesis and software development are presented.
SIGNIFICANT RESULTS
• The work reported here established the total dynamic system approach to the analysis of the digital flight control problem.
• The chief benefit of the program is to provide software for rapid evalua- tion of system performance.
• A sufficiency rule for sample rate selection is developed (the first quadrant rule).
Requirements for digital computation of control laws are established for sys- tems with bandwidths of 6, 12, 20, and 25 Hz. Future growth for control configured vehicles requires this broad bandwidth closed loop controllability.
Table 18 shows sufficient sample rate requirements as functions of system bandwidths using this rule.
Sufficient coefficient word length requirement of 16 bits is established for a digital controlle: . Maximum allowable computational delay is established to be less than one-fourth the sample time.
280
Table 18. Sufficient Sample Rate Requirements vs. Bandwidth
Significant Mode in Vehicle (Hz) (rad/sec) |
Sufficient Sample Rate (samples/second) |
|
6 |
37.5 |
25 |
12 |
75 |
50 |
20 |
125 |
80 |
25 |
160 |
100 |
RECOMMENDATIONS FOR FUTURE ANALYSIS WORK
• For large sample times the response model at sample times is not sufficient to assess the performance. The intersample performance is needed.
• Development of average performance measures is needed for systems with large sample times. Phase and gain margins and frequency responses have to be carefully defined since a digital control system is essentially a time varying system with periodic coefficients, and for large sample times the response to a sinusoidal input is periodic but not necessarily sinusoidal.
• Loop breaking (one at a time) for the phase and gain margin analysis should be put on a good foundation for multiloop systems.
RECOMMENDATIONS FOR FUTURE SOFTWARE DEVELOPMENT WORK
• The existing analysis and corresponding DIGIKON programs should be extended to include the methodology of direct digital design.
• An image w-plane (i.e. , z-plane) root calculation capability should be added to DIGIKON.
• Intersample modeling for frequency response should be developed,
• A hybrid transfer function capability should be added to the DIGIKON frequency response program. This would allow the computation of frequency response for a hybrid system consisting of both continuous and digital quadruples.
• A composite input capability [see Equation (320)] should be added to the DIGIKON frequency response program.
1
3
L
281
:»
CONCLUSIONS
A larg£ -scale system software for the analysis of digital flight control systems is developed in this study. The programs which implement the theoretical models are documented with the user in mind. A parametric study was per- formed on a typical tactical fighter aircraft, the F-4, to determine computa- tional requirements.
1
REFERENCES
1. J. F. Kaiser, "Some Practical Considerations in the Realization of Linear Digital Filter," 3rd Allerton Conference, Oct. 20-22, 1965.
2. J. B. Knowles and E. M. Olcayto, "Coefficient Accuracy and Digital Filter Response, " IEEE Trans, on Circuit Theory, Vol. CT-15, No. 1, March 1968.
3. J. B. Knowles, and R. Edwards, "Effect of a Finite-Word- Length Com- puter in a Sampled-Data Feedback System," Proc. IEE, Vol. 112,
No. 6, June 1965.
4. B. Gold, andC.M. Rader, "Effects of Quantization Noise in Digital Filters, " Proc. Spring Joint Computer Conference, 1965.
5. W. R. Bennett, "Spectra of Quantized Signals, " Bell System Technical Journal, Vol. 27, pp. 446-472, July 1948.
6. R.K. Cavin, "Quantization Error Bounds for Hybrid Control Systems," PhD diss irtation. Auburn University, Auburn, Ala,, June 1968.
7. Technical Report AFFDL-TR-71-20, Supplement 2, "Survivable Flight Control System Interim Report No. 1, Studies, Analysis, and Approach. " McDonnell Aircraft Company, May 1971.
8. A.F. Konar and M. D. Ward, "Development of Weapon Delivery Models and Analysis Programs, " Technical Report AFFDL-TR-71-123, Vol. I.
283
appendix a
SIMKTC -- MODEL FOR F-4 CONTROLLER VIA TRANSFER
FUNCTION INPUT
The F-4 longitudinal controller presented in the fly-by-wire report AFFDL-TR- 71-20, Supplement 2, is used in the following as a demonstration example for the quadruple generation via transfer function input.
Figure A1 shows the controller transfer function block diagram. Figure A2 shows controller "data" block diagram as well as input and output variables There are six subsystems. Each subsystem is of first order, except sub- system 2 which is of second order. The equations describing this system are
given below. These equations are implemented in subroutine SIMKTC. Figure A3 shows its program listing.
x(N) = A(n) x(n) + B(n) u.(n) n = 1, 2 6
ri(N) = C(n) x(n) + D(n) u.(n) n = 1, 2, . . . , 6
u.(l) = r.(2)
^(2) = r.(3) ui(3) = r.(4)
ui<4) * KVKP C-Kcs r.(5) + Kq u(3) + r.(6)] ui(5) = 2 u(l )
ui(6) = u(2)
(Al)
(A2)
(A3) (A4) (A 5) (A6) (A 7) (A 8)
r(l) = r.(l)
(A9)
Preceding page blank 235
Diagram of a Controller Transfer Function
SUBROUTINE S1NKTC
COMMON V(41 ) *V(7I) *NX*MY*NR»NU* INIT* IFLAO*MODE*F(41*70> *T*IFC COMMON/S 3/ ROOT (3*10)*X(3*10)*RI(3*10)*UI(3*10)*U(7) *NNX ( 10) * 1NNR<10)*NNU(10).NMAX*ISO*ISOMAX*TPS*IRRINT
CONMON/TF/ AT (3*3*6) *BT (3*1*6) »CT ( 1 *3*4) *DT (1*1*6) *M*PRI
1 NT (3*3)* hS (2*3*6)
DIMENSION 9(1)
REAL KV*KF»KCS*KO*KNZ
C initialize
IF(INIT.NC.O) GO TO* 100 READ(5*400) KV»KNZ*KO*KCS 400 FORMAT (10F8 .3)
CALL 0AInTAB(IFC*KF)
WRITE (9*765) KV«KNEtKQ*KCStKF
765 FORMAT <lX«3HKV**£12.5*4HKNZa*E12.5*3HK0**E12.S*4HKCSa,E12. 5.
1 3MKF**El2.5)
READ(S* 101 ) NMAX* (NNX (N) *N»1 tNMAX)
101 FORMAT ( 12* 1012)
NXaO
DO 102 Nal*NMAX NX*NX«NNX(N)
NNU(N)*1
102 NNR(N)-1 NR«1 NU*3 NY.12
C READ IN HS MATRICES DO 600 Nal.NMAX NN1*NNX(N)*1
READ(S*60l ) ((HS(I»J*N)*Ial*2)*Jal,NNl)
601 FORMAT (5El4«B)
400 CONTINUE C PRINT HS MATRICES
DO 250 Nal.NMAX WRITE (9*403) N
403 FORMAT ( 1X*3HHS( « 12* 1H) )
NNlaNNX (N) ♦ 1 DO 402 1*1*2 DO 402 J»1 ,NN1 402 MlNT(I*J)aHS(I*J*N)
CALL MPRS(RRINT*3*3*2*NN1*0»4H )
C COMPUTE QUADRUPLES FOR ALL BLOCKS CALL TRANSK 250 CONTINUE RETURN
100 CONTINUE
11*0
C COMPUTE SUBSYSTEM STATES XOOT(N)-AN«XN «bn*un DO 251 N«1*NMAX MX-NNX(N)
DO 200 1*1. MX 11*12*1 VdIIM.0 NUXaNNU(NN)
Figure A3. Program Luting for Subroutine SIMTC
288
201
00201 J-1»NUX
V<in«V(Il)*8T(I.J,N)»UlCJ.N»
DO 200 J-l.MX
200 V C 1 1 > »V < 1 1 > ♦ AT (I»J*N)*X(JtN)
251 CONTINUE
C COMPUTE INTERNAL outputs rin-cn«xn«dn*un DO 350 N>1»NMAX MXaNNR(N)
00 300 I«1 f MX II«1!»1 V<II)-0.0 KX1«NNX(N)
DO 301 J«ltMXl
V(II)»V(II)*CT(ItJ»N)#X(J»N)
NXlaNNU(N)
00 300 Jal*NXl
V(II)aV(II>*OT(I,JtN)*UI<J,N)
CONTINUE C INTERCONNECTION equations
V(lA)«Rl(lf2>
V(1S)-RI(1,3)
V(16)aRI(l,4)
v! 18)15(1 r# ‘ (‘2*#,<CS,*Rl 11 **> •K0*UI3I ♦KNZ-RJ ( 1 ,6))
V<19)-U(2>
C external response
V (20) «RI ( 1 « 1 )
RETURN
ENO
301
309
350
Figure A3. Program Listing for Subroutine SIMTC (Concluded)
289
-
The subsystem quadruples A(n), B(n), C(n), D(n) for n ■ 1, 2, .... 6 are generated by the subroutine TRANSK, using transfer function input data. Equations (A3) through (A9) are called the interconnection equations. They can be replaced by Equations (A10) and (All) if the system interconnection is specified by the interconnection quadruple (PC, QC, RC, SC).
u.(N) = P r (n)+Q u.(n) n = 1, 6 (A10)
A v 1 v 1
R(l) = Rcr.(n) + Scu.(n) (All)
These can be implemented as shown in Figure A4. The nonzero elements of the interconnection quadruple for this example are as follows.
Pc0.2) Pc(2.3) Pc(3.4) Pc<4, 5) Pc(4.6)
= 1 = l = 1
= K K_, (-2) K v F cs
= K K_, K v F nz
Qc(4, 3) - KvKFKq Qc<5. 1) = l Qc(6,2) = l
Rc(l. 1) = l
The output of Subroutine SIMKTC was compared with the output of Subroutine SIMKC. Complete agreement was noted. The interconnection quadruple version of SIMKTC was not implemented in the DIGIKON system.
Complete documentation of SIMKC and SIMKTC are given in Volume II of this report.
290
c
INTERNAL INPUTS
II = NX + NMAX DO 400 I = l. NMAX II = II + 1 V(H) = 0
DO 401 J = l, NU
401 V (II) = V (II) + QC (I. J) * U(J)
DO 400 J = 1, NMAX
400 V (II) - V (II) + PC (I, J) * RI (1,J)
C EXTERNAL RESPONSE DO 500 I = 1, NR II = II + 1 V (II) = 0
DO 501 J = 1, NU
501 V (II) = V (II) + SC (I, J) * U (J)
DO 500 J = l, NMAX
500 V (II) = V (II) + RC (I, J) * RI (1. J)
Figure A4. Interconnection Equations Via Interconnection Quadruple
291
APPENDIX B
rr.
STATE MATRIX APPROACH TO ROUND-OFF NOISE ANALYSIS
OF DIGITAL FILTERS
ABSTRACT
Methods for round-off noise analysis of digital filters are briefly presented.
Results of modern control theory are applied to this problem by viewing digital filter as a discrete -time system and treating fine quantization as a random noise driver, acting at various points.
The structural sensitivity of the output noise to various realization schemes is briefly discussed for efficient digital filter mechanization.
The state matrix technique presented here facilitates the noise investigations of digital filters used in open or closed loop operations by a digital computer. This technique is straightforward and can be used in a paper and pencil analysis.
1. INTRODUCTION
The advent of integrated circuit technology has greatly Increased the possibility of constructing digital micro-processors thai can compete with analog hardware in control and communication systems relative to economy, size and reliability considerations.
A linear digital filter is a discrete-time system. Its transfer function is f implemented either by programming on a digital computer or by realizing
t
\ _
292
it as a digital network. Emphasis is given here to the design of special pur- pose hardware to realize a digital filter as a system component (i. e. , micro- processor).
In performing the hardware design there is great flexibility available. The designer must specify not only the sampling rate to be used but also the number of bits to be used to represent the slgnal.to represent the filter coefficients and to be used in the product accumulation, type of arithmetic to bo used and the procedure for rounding or truncating the results of arithmetic operations. All these design parameters have a direct effect on the noise .sensitivities and complexity of the filter mechanization.
Several numerical problems can arise in the design and utilization of digital filters.
Computational errors (i. e. , digital noise) are introduced within the filter due to (i) truncation of filter coefficients, (li) quantization of input data, and (iii) rounding-off the result of multiplications (Refs. 1-4).
In this work, a method is presented for the analysis of errors due to quanti- zation and round-off by using the State Matrix approach.
Fine quantization or rounding may be treated as additive random noise (Ref. 5), Such additive noise is nearly white, with a mean- squared value of q / 1 2 and mean zero, where q is the quantization level, The main problem of the analysis is the development of an expression for the mean-squared output of an arbitrary filter excited by a white noise source entering into the various points of the filter (i. e. , after quantizers and multipliers). There are two points of major interest in the analysis:(i) investigation of the transient behavior of the mean-squared vglue of output noise over a discrete-time interval, (ii) determining the steady-state mean-squared value of output noise, when it exists.
*1 li'» I '• I ill I lit
293
tm
The first case is of practical interest when the noise built-up in digital resonators and sine -wave generators are studied. The second case is of practical interest for determining the steady-state performance of asymptoti- cally stable filters.
The realization of digital filters is discussed in Section 2. A Realization Theor- em is stated for establishing equivalence between a digital system and its ideal implementation. Frobenius Input, Frobenius Output and Jordan Systems arc illustrated realizing the same transfer function. In Section 3, methods for noise response analysis of the various realizations are discussed,
2. REALIZATION AND STATE VECTOR ASSOCIATION OF DIGITAL FILTERS
The essence of the realization is this: given a discrete time system L^ characterized by, say, an input-output relation of the form
D(z) y - N(z) u (Bt)
where
D(z) ■ aQ+ajZ 1+....+anz n N(z) » b0+bjZ *+,... +bnz n
are difference operators, one constructs an equivalent system L2 in the form of an interconnection of adders, scalors, and delayors,
L2 is said to be equivalent to L^ if (i) L^ and L2 are zero-state equivalent, and (ii) Lj and L2 are zero -input equivalent. When these conditions are fulfilled, L2 is also said to be a realization of Lj. The establishment of the equivalence between Lj and L2 is facilitated by the following Realization Theorem:
294
"Let L1 be a discrete-time system of order n, characterised by an input-output relation of the form
D(z) y * N(z) u
in which the polynomials D(z) and N(z) do not have common factors. Let L2 be an interconnection of scalors, adders, and delayors which is zero-state equivalent to Lj and which contains exactly n delayors. Then L2 ia equivalent toLj."
Now consider a discrete-time system characterized by Equation (Bl). If D(z) and N(z) have no factors in common, one can construct Lg simply by synthesizing the transfer function
u/7\ m N( z)
H(z) T3TZ7
through the usual techniques of circuit theory (which yields a system that is zero-state equivalent to L j). Subsequently, one invokes the realization theorem to establish that L2 is equivalent to L^. Then one associates a state vector x(kT) with L2 by assigning a component of x(kT) to the output of each delayor. Since Lj and Lg are equivalent systems, x(kT) qualifies as a state vector for Lj and the state equations of Lg may also be regarded as being the state equations of Lj,
In the following, realization techniques will be applied to digital filters for purpose of illustration:
295
2. 1 Discrete-Time Systems Without Numerator Dynamics
These types of systems are characterized by the transfer function
b»
Lj = H(z) ■
ITT . a0 i 0
a0+alz +,,,+#nz
It is easy to verify that the network Lj shown in Figure B1 has the same transfer function and hence is zero-state equivalent to Lj.
U -i I -i |
-1 , -1 |
|
T\Jo/ |
liJTil |
|
\ |
x2(k) |
Figure Bl. Realization of — —
a0 +
Frobenius System
Bq + SjZ 1 + ...+anz n
as an Input -
Furthermore, L2 has exactly n delayors. Therefore, by the realization Theorem, L2 is equivalent to Lj. Now a state vector x(kT) can be associated with Lg by assigning a component of x(kT) to the output of each delayor. The defining relations for x(kT) are:
Xj(kT) ■ y C (k-n) T]
x(kT) ■ y C(k-l) T] n
An alternate realization of Lj is shown in Figure B2.
iKk)
Frobenius System
The defining relations for x<kT) are:
x^k+l) * - aiy<k) +X2(k) (B4)
Vl(k+1) ■ " Vl y(k) +xn(k)
xn(k+l) ■ - any(k)
with
y(k) - -L [ Xl(k) + b0 u(k)] (B5)
2, 2 Discrete Systems With Numerator Dynamics
Let Lj be a discrete-time system with numerator dynamics characterized by the transfer function
297
b«+b,z +,
Assume that the numerator and the denominator do not have common factors. Clearly the network shown in Figure B3 realizes this transfer function [ i. e. , has transfer function H(z)].
Figure B3. Realization of
aQ + a^z +»**+*n*
as an Output-
Frobenius System
Since Lg has exactly n delayors, Lg is equivalent to L^, An alternate realization of L, is shown in Figure B4.
Figure B4. Realization of
V*iI‘' + -"+V’n
as an Input
Frobenius System
•Tv
2. 3 Realization aa a Jordan System
Let Lj be a discrete system characterized by the transfer function
N(z) (B7)
H(z)
unjz'1) (i4x2*'l)....(1nnz’1)
in which, V. 1 = 1,2,..., n are distinct and no (1+X.z ) is a factor of N(z).
Furthermore, let the partial-fraction expansion of the transfer function H(z) be
H(z) ^-r +...+ ~tt ** (B8)
ri rn
rr +* • • + rr
(l+x^ ) d+xnz >
Then the system Lg shown in Figure B5 is a realization of Lj and the vector x(kT) defined by its components
x^k+l) ■ X^k) x^k) + u(k), i«l, 2...n qualifies as a state vector for Lj u(k )
(B9)
y(k)
299
iJtoararnkiil'iifilr'ii^yii'iYlili^^iiirnVii-iili
i ** ■ flam ■*-
Figure B6. Realization of H(z) with Multiple Poles as a Jordan System
300
This finishes the discussion on the realization and state vector association of digital filters. In the next section, methods will be given for analyzing the noise behavior of various realizations.
3. THE NOISE BEHAVIOR OF VARIOUS REALIZATIONS
The least upper error bound of the maximum noise output is developed in Reference 6, Practically, this bound is very conservative and almost use- less. An estimate of the mean square output noise of a digital filter is given in Reference 3, utilizing the maximum gain of the frequency response of its discrete transfer function. In Reference 4 the mean square value of output noise is derived via the transfer function approach.
In the following, first this approach is briefly presented for completeness, then the state matrix technique is developed. Finally, the steps in the analysis are outlined for a notch filter.
3. 1 Development of Mean-Squared Response Equation via Transfer Function
Let [w(kT)] be a noise sequence with known statistical properties applied to a digital filter, Let H(z) be the transfer function between the output of the filter and the node where noise is injected, and y(kT) be the resulting output noise sequence as shown in Figure B7.
w(kT)
Figure B7. Random Noise Applied to a Filter
301
— - -
If the filter is initially at rest and the input noise w(kT) is zero for k <0 then the output is given by the convolution sum
y(kT)
h(mT) w(kT-mT)
where h(mT) is the inverse-z-transform of H(z) (i.e. . impulse response). Squaring Equation (B12) yields
y2(kT)
h(mT) w(kT-mT) ) h(*T) w(kT-tT)
k k
ZZ
m»o -t»o
h(rnT) hfrtT) w(kT-mT) w(kT-tT)
(B13)
Now if w{k - ) is a random variable with zero-mean and variance o2 and if w(kT) is independent from sample to sample, the expected value of Equation (B13) becomes
E C y tkT)]
k k
zz
m*o <>■0
h(mT) h(£T) EC w(kT-mT) w(kT~tT)J
y2(kT) » ) h2(mT)ff2
(B14)
The steady-state mean-squared value of y(kT), if it exists, can be obtained by letting k approach infinity.
If only the steady-state value of output mean-squared noise is of interest it can be found without computing the impulse response of the filter. This classical result is demonstrated by observing that
h(mT) - Ti-r £ H(z) zm_1dz Aversion Theorem) (B15)
Substituting Equation (B15) into Equation (B14) and interchanging the order of the summation and the integration yields
2
y2 ° HU) z 1 dz ^ h(mT)
_m
(B 16)
m»o
but
*’ ' I h"
HU *) * ) h(mT) zm (by the definition of the z-transform) (B17)
m*o
so that
y * ft H(z) H(z *) z 1 dz
(B 18)
where the contour integration is taken along the unit circle. To illustrate
the technique, consider a first-order filter characterized by the transfer function
H(z) »
1+ajZ
(B19)
303
<S!
S
Let w(k) represent combined quantization and round off noise, with mean zero 2
and variance a . Figure B8 illustrates the corresponding equivalent system.
w(k)
k
.-i
y<k)
Figure B8. First-Order Digital Filter Driven by a Noise
From Equation (B18)
2 _ _2 r 1 a z"1 1
y ' 0 7^
dz]
(B20)
and by applying the Residue Theorem to the contour integral yields steady-state mean-squared output noise
1 -a
T
(B21)
One notes that is the pole of H(z), As approaches the unit circle, the mean-squared value of the steady-state noise increases without bound.
3. 2 Development of Mean-Squared Response Equation via State Matrix Approach
Let x(kT) be a state-vector associated with a realization of a digital filter as described in Section 2. Then, the evolution of the state and output due to a noise sequence [w(kT)] is described by
x[(krl)T] = F x(kT) + G w(kT) y(kT) = h'x(kT) + d' w(kT)
where F, G, h, and d are transition, noise input, noise output, and noise transmission matrixes with dimensions nxn, nxr, nxl, and rxl, respectively.
Now let X(kT) be an nxn matrix (i. e. , state matrix of the mean-squared response) defined by
X(kT) = E { x(kT) x'(kT)} (B23)
where prime indicates the transpose. Similarly let
W(kT) = E {w(kT) w'(kT)] (B24)
Then, from Equations (B22), (B23), and (B24) it follows that X(k + 1) = FX(k) F'+GW(k)G'
(B25)
y2 = h'X(k) h + d' W(k) d
in which T is dropped for simplicity in writing. The set of equations defined by Equation (B25) completely specifies the evolution of the mean-squared value of output noise on a discrete-time interval
{ kT } ; k=o, 1, . .
305
Since round-off and quantization noises are assumed to be stationary, it follows that output noise has a steady-state mean-squared value If the filter is asympto- tically stable. In this case the solution of the equation
X = FXF' + GWG'
for X and then evaluation of y^ =h' Xh + d' Wd yields the steady state mean-squared response.
(B26)
(B27)
For purposes of illustration, consider a second-order filter characterized by
/
1
H(z) =
_ i o o i r < 1 1 - (2 r cos 0T) z + r z~Z
(B28)
Let w(kT) be the combined quantization and round-off noises with a zero-mean
and variance a . Figure B9 illustrates the corresponding equivalent Frobenius- Input System.
w(k)
-ay(k)
L_ |
-1 |
x^fk) |
-1 1 |
||
A |
z |
ill |
2 r cos I
x2(k)
Figure B9. Second-Order Digital Filter Driven by a Noise
306
l.BIM
The equivalent system is described by
x(k + 1) = F x(k) + g w<k)
y(k) = h' x(k) + d w(k) = x2 <k + 1)
where
F =
(B29)
r i — ■ ■ |
i |
. g * |
O' |
h = |
2 •r |
L" r |
2r cos j3T_ |
.1 . |
$ n |
2 r cos 0T. |
. d = 1
Substituting Equation (B29) into Equation (B26) yields the state equation of the mean- sauared response.
X1I <k+ ') = x22(k)
x12(k+l) = -r x12(k) + 2 r cos 0T x^ik) + o‘
(B30)
(B31)
x22(k+1) = r4xn<k)-4r3cos0Tx12(k) + 4r2cos^Tx22(k)+a2
One notes that the poles of H(z) in Equation (B28) are N.2 = r (coa @T ± sin 0T)
Thus With r < 1, the filter is asymptotically stable. In this case the steady-state response is given by
X (k + 1) = x(k) = X
(B32)
307
So that
X11 |
= X22 = y |
|
X12 |
2 = -r x12 + 2 r cos 0T x22 |
(B33) |
X22 |
= r4 xji - 4r3 cos 0T xJ2 + 4r2 cos 0T x22 + o2 |
Solving the set of equations defined by Equation (B33) and after some algebra, one obtains
1 + r
1 + r* - 2r2 cob 2 0T
x
12
2r cos AT
(B34)
Clearly, when r approaches 1, y grows without bound as expected.
It should be pointed out that in the example given above. Equation (B26) has been solved analytically since the filter has been of relatively low order.
For practical systems requiring digital filters of higher order, for instance of order ten, the analytical solution would be extremely tedious. As is indi- cated below, state matrix formulation of this problem facilitates the solution by a digital computer very efficiently.
It should also be pointed out that in many cases, digital filters are used in an open loop fashion. For this class of applications the mean square output noise of the filter itself is a meaningful parameter for measuring the system per- formance. However, if the filter is used in a closed loop, for instance in an automatic flight control system, then obviously the study of open-loop perfor- mance of a filter alone is insufficient to predict the overall system performance.
303
In this case the state matrix approach becomes a very convenient tool, since a discrete model of the overall system can readily be developed in the form of Equation B25.
Relation (B25) is known as the discrete" Lvapunov Equation." It is also referred to as "the discrete state covariance equation. " It can easily be verified that the partial sum generated by the iterative solution of Equation (B25) satisfied the following recurrence relation
«k-l 2k-l
Sk = F Sk-1 F + Sk-1' k = 1*2* —
with SQ = GWG'
and X = lim
k -» ®
2
Once X is computed as indicated above, the output mean square noise y can easily be obtained from Equation (B27). Clearly subroutines already developed for control system design purposes can readily be utilized for this calculation also.
3. 3 Structural Sensitivity
To explain the method for studying the effect of various realization structures upon the mean-squared output noise response, consider an analog notch-filter characterized by the transfer function
G(s)
p2 x 2
s + a (s + a)2
(B35)
309
E.fflBPU I'M.HHJ I il l.l l I I I W. IBI^I W. J > iJW -D jli^li) . 'I.
i»j ?»*»?■
wmj* MIX! ■»*,«* »»»l! J.HM 1"
Letting
-1
f1 .Iitl
2 l'-.*1
yields the following digitized transfer function
HU) = 3-tl£ «r‘ ♦ .*1 t'2
<i +c * r
or
= i±i& y + y'2,,
1 + 2C z -1 +C2 z"2
(B86)
(BS7)
where
C
= 2yTl
ti - *
¥
+ 1
^T)
and y
aT
2
(338)
It can easily be shown that Equation (B37) can also be written as
H(z)
1
^ +
r
+ d
(1 + C * > (1 +C * )
(B39)
where rj = r\ - 2 + d, rg = 2 - 2 d, with d (B40)
Let T be such that no truncation occurs in the representation of coefficients appearing in Equations (B37) and (B39). Further assume that the result of each multiplication is rounded, then summed. With these assumptions, Fig ures BIO and Bll correspond to the noise models of the filter based on Equation (B37); and Figure B12, based on Equation (B39).
310
Figure BIO. Noise Model of a Digital Filter Realized as an Output- Frobenius System
It should be noted that these are not the only possible realizations. The reader can add to the list, other alternate realizations. However, the important point to remember is that the equivalency of these realizations is valid only when noise is not present. Obviously, each network illustrated above yields
a different mean-squared output noise value for the same quantization and round-off errors.
The variation of the mean-squared output noise of a digital filter with respect to its realization schemes is termed, The Structural Sensitivity.
Clearly, the best dynamic realization in the sense of. yielding the least mean- squared output noise depends upon the coefficients of the filter transfer func- tion as well as the input quantization level and the word length of the computations.
*
311
312
To find the best dynamic realization, one first obtains the matrixes F, G, h, and d for each possible realization. Then the mean- squared output noise is computed as described in Section 2 for given quantisation level and word length.
It should be remarked that in the actual design process, not only the dynamic performance, but also hardware aspects of the filter are considered as well. Therefore, the ability to predict analytically the noise behavior of various realizations is of practical importance in the tradeoff studies of digital filter mechanization.
4. CONCLUSIONS
A convenient technique for the noise analysis of digital filters by a digital computer is presented. The technique is applicable to both open and closed- loop systems.
The concept of structural sensitivity of the output noise to various realization schemes is briefly discussed for an efficient filter mechanization.
313
•
■ —
APPENDIX C
MODELING OF F-4 LONGITUDINAL CONTROL SYSTEM WITH TIME DELAY
A reduced model was developed for the parametric study of computational time delay effects. In this model the actuator and the gust dynamics are modified (a +hird order actuator and a second order gust filter). The same model with minor modifications is used in the simulation tests.
Figures Cl and C2 represent the block diagram and state equations for the actuator, and Figures C3 and C4 represent the block diagram and state equations for the vehicle. Figures C5 through C7 contain the program listings of the subroutines SIMKA, SIMKV and SIMKP respectively corresponding to the new models.
The subroutine HSIMK is used to introduce time delay into the overall system model. The subroutine SIMK presented previously combines the quadruples without time delay. It is used here for checking the outputs of the subroutine HSIMK for zero time delay. The amount of time delay 0, T/4, T/2, and T, where T = sample time, is specified by an input timing sequence ISIMK (ISQ),
ISQ =1, .... ISQMAX. This is read in the subroutine STAMK4. For each value of ISIMK (ISQ), the corresponding subsystem is updated in the subroutine HSIMK as explained in Section III.
The program listings of the subroutines STAMK4 and HSIMK are given in Figures C8 and C9 respectively.
l
Reduced Block Diagram of Actuator
Differential Equations
X(l) = KMR * Y(l)
X(2) = BETA 2 * BETA 3 * Y(2)
X(3) = Y(3)
Summing Point Equations
Y(l) = LINKG1 * X(2) - C ST A BA * X(l)
Y (2) = BETA1 * U(l) - Y(3)
Y(3) = BETA 1 BDMOD2 * X(l) + CGOK * BDMOD1 * CVPIN *X(2) - X(3) )
Response Equations
R(l) = LINKG2 * CRPD * X(l)
R(2) = LINKG2 * CRPD * X(l)
R(3) * KMR * LINKG2 * CRPD * (LINKG1 * X(2) - CSTABA * X(l))
Values of the Parameters
BETA1 = . 37 BETA2 = 57. 6 BETA 3 = . 408 BDMOD1 = 1.25 BDMOD2 = 1. 17 LINKG1 = 1.372 LINKG2 = 2. 865 KMR = 133.
CSTABA = 1. /7. 128 CRPD = 1. / 57. 3 CVPIN = 14.
CGOK = . 296
i
\
'
Figure C2. Reduced Actuator Equations
316
J
r
Differential Equations
V» 1 <> + ze> xv(2) + z» Yv(1) + zr, Xv(4)
+ \ Xv(3) + \ xv(6> + Z^ xv(5> + % V8)
+ \ Xy<7> + z6 Uv(3) + Z-g Uy<2) + Zg Uv(t)
o
Xy(2) = Yv(2) + Ma Yv(l) + Mj Xy(2) + M- + Xy<4)
+ \ Xv(3> + % xv(6) + Mf,„ Xv<5> + xv<8>
+ Mn, XV(7) + M6 Uv<3) + M« Uv(2) + M« UV(U o
Xy(3) = Xy(4)
Xv(4) = F» Yv(1> + FS Xv(2) + Fr, Xv(4) + Fn Xv<3> + F* Xv,6)
112
+ Fn2 xv<5) + % xv(8> + Fruj xv<7> + F« uv<3>
+ f5 uv<2) + Fa uv(1)
Xv(5) » Xv(6)
Xv(6)=G«yv(l) + G8Xl<2> + G* Xv<4)+G„ Xv(3) + GT| Xy<6>
112
+ % X2<5) + % Xv(3> + % Xv<7> + GJ Uy(3)
+ Uv<2> + G« Uy(l)
Figure C4. Vehicle Equations
Ml IIIIWIP'II
L
wmf.i ui'«"'^«i' 'i-i "iiiipu
Xy(7) = Xv(8)
X (8) = H Y (!) + Hi X (2) + Hi X (4) + H„ X (3) + Hi X (6) v a v 0 j* ri, v *1, v T)„ v
'1
'1
+ H X (5) + H* X (8) + H X (7) + HVU (3) ^2 v ri3 v n v 0 v'
+ H« U (2) + H« U (1) 0 v 0 v
1
Xv(9) =^- [-Xv(9) - Xv(10) + ( V3T cr/UQ) Uy(4)]
W
Xv(10)=^r [-Xv(10)+ ((V3 - 1) VTror/U0)Uv(4)] w
Summing Point Equations
Y (1) = X (9) +X (1)
V V V
Yy(2) =Xv(9)+Xv(l)
Yy(3) = Xy(2) + Xy(4) + (ac^/a5) Xy(6) + (acp3/8^) Xy(8)
Yv(4)"^(xv(2)“iv(l))
Y (5) = Y (4) + (L /3?.2)X.(2)
V V A JL
Yv(6) * Yv(5> - SO- Itci V4) + *2 V6> + ®3 V8)1
Respor Equations
r (1) = Y (1)
y V
r (2) = Y (6) v v
Figure C4. Vehicle Equations (Concluded)
319
i.kdjLt-ii.**, * «>» ^ Ain ifru
oor» Don nno
surpohtinf stmxa C SI*** R*00 VERSION
C simulation equations mo r-u aftijator
0
COMMO*' V <4»1 ) • W ( 70 1 .NX.NY.N».N'it INTT. JfLAG.MOOE.F Ul .70) .T. I PC OfMFNcTON X ( 7 ) . XOOT ( 7 ) « Y { 3)»"<1)
OPAL <MR.LINKG1 .L IMKGS
POUT V M.FNCF (* DOT ( 1 ) .W( 1 ) ) . (v <1).*( 4)).<X (|).w< 7)).
HU (1 i .*( 10) )
IP { INTT.NF.0) GO TO loo
N X = 3 NYs7 Nil* 1
NR = 7 RFTlJR’
100 PONT T Nl IF
PFTA1=.77 qFTA?rG7.ft PFTA1=.40R l. IMKGisl .37?
L TN*G? = ?.flf>S KMRs 1 73. rSTAOA=l./7.1?"
CRPO=l . /57 . 7 rVPIN=l4.
ROMOOI -J1.2S «nMon?=i . 1 7 TOOK s. 300
0 IFFFRFNT » AL EQUATIONS
V ( 1 ) s'Mg.yi 1 )
V (?) spFTA?*RFTA3*Y (?) VI3HVI1I
summing point equations
V( A) =LINK01»X <?)-CSTAHA*X ( 1 )
V(S)*MFTA1*IH1 !-Y(7)
vos) »bftai«romoo?*»x ( i ) ♦CGnK#H0Mnni*r.vpiN*x i?)-x O) OUTPUT F0MATI0NS
V( 7)=LINKG2*CPP0*X(1)
V( 8)«LlNKfi?*C»P0*X00T(l>
_V( 9)«KMR«LINKG?*CPPD*(LlNKr,l*X00T(2)-CSTABA*xn0T( 1 ) > WTORm FN0
Figure C5. Redvced Actuator Program Listing
SUAftO'iTIMF S|M*V
S I MLK_ ATIO*1 FGUATTONS foo r-^vrntCiF SfMKV 6*00 VFOSTON
COMMON VU1 » .M( 7fM .NX.Nr .NO.Nn, INI T • jrt AG»**OOF .niHF lft| • 70 w T. |FC OIMfnmon 7l1!i^i|?i.ri||i.r.i||i,Mi||i n?MFNC?ON RnOTim.V(M.*l)ni.iMftl.PMl MMFNMON ARIJF | AM l
FQUIVALFNCF I ROOT ( T I • 41 1 ) ) . ( Y i I ) «M 1 1 1 t) • l x ( ) I • n>U|VAlCNCF iiH'ir(c),7iii>,<4oiir(iM,4{iii,i4K’'ri?»i.ri|n.
1 ( arijf < iQ) .ou j \ . ( iRiif mo ) «m( i ) i , ( Amir imi .oomi | > •
? <A«UF (ft? I ,r)PMl?i • lARUF 1631 «0Ph| n • ( AR JF (6ft I *pm| | ) ,
3 I ARUF ( ftS ) • on I ? i • ( 6RUF (66) .PhI 1) , < AbUF (67) •LFNOTh) .
ft (ARuri**), ips 7>
ofal *>.L* •LFNOTh*L‘- I F < INtT.NF.rt) r,0 To ion
c
C INITIAL t7c
c
CALL °ATA (ARUF. 11 NR * l 0
N V s *•
NO * **
Nil r
L* * (LFNC.Tm - 7 7 . -> » / l?.f>
r
C V I ND F I l T r p INPUT
c
LM*1 7*0.
TM*L M /I i°SZ stoma 2 7.n
pfmin^ 7
RFT*JP‘i 100 CONTI‘inr
c
C OIF rfOFNT T AL EQUATIONS C
viii * 7 ( i » *v ( i ) • 1 1 .o. y i > 1 1 •* i ?\ ♦ /(3»»*»fci •
1* 7 ( S » • ■ ( 6 ) • M M ■ I ft l . > f 7 . • i f A » . 7 ( * 1 • > ( 7 \ • 7 ( O | <• j ( 1 |
?• 7 M* 1 *11171 • 7(11 »*'ll 1 >
V(?» » M I 1 » • V I 1 ) • u(?|»V(>) ♦ M ( 3 | • « | > > • ••U.IMUt
1* M ( S * • X M I ♦ “ 1*1 "X (ft l . u f 7 l ° X IS) . *(«)•■ I «1 • M|Q»*R(t»
?• Mipi»iini . 'um^iiu ♦ mi|0)°mii
V M> = X < A )
v<ft» * r 1 1 > • r 1 1 * ♦ F(?i*i(5i • Fin •xK.i ♦ rui**ni
1 ♦ f ( S * • r ( 6 1 ♦ r c •* i i * i ft » • f i 7 t • x ( ft i • r r m «* 1 1 7 t ♦ F ( o l «* j i w
?♦ F (l'M«U(?l • F f | J Mm | >
V IS> * R (ft 1
VIM * O (I I «* i I I » • O ( 7 ) ♦* r ( •> I . r M >**(<■ I • n ( ft I • R M i
| • O I S * • R 1 6 l • o I ft i *• ■ ( ft i • O ( 7 « • i I A i • r, | H > ® * ( 7 » • r. | Q » • i f 1 1
?• r,| 1 * j •*!(?) • rmimili
V ( '» I r * ( A )
VlAl = HIIIMlIl • -*(?»«»»(•; • m M ) • I ( <* I • •' (ft I *1(31
1* M ( S * • R ( 0 > • w{Mnj(C| ♦ h(7i«I (PI ♦ H I H » * ■ I ? » • u|Q|«uni
?• h(1 m*hm» • h( | i i-m 1 1 V(Q|«(-R(Q)>V(|ni.coPTi‘i.n»TMi*S10MA*jift)/>lPS7)/T4
V ( 1 ft I * I — R 1 10) • O OP T (TWt".7"*?«»Sl OM A * 1 1 ( u 1 / JP S / 1 / T ••
r
C COMPUTF r FOUATTONft
r
Villi - * ( Q > • * f l * v ( i ?) = root < i i
villi = km* • ''uwi i <» r < <. ) * • noHii»i m
VUft) = (UPS7/1?. ?) « I X ( •> ) -R' m I 1 » »
vilSi = viftj ♦ ii i/i5,;i#in*n?i
V ( | 6 1 = MS» - (OMl i^xDOT fft» .PHi?*x0OT (ftl .PHI 3*xH0T (A) 1 /!?. ?
C
c OATF AJO ArrFlFOATfON O'jToiit
r
V 1 1 *7 1 = Y(3)
V I 1 A) = V (01 OFTUP‘1
F NO
f
%
•1
i
vigure C6. VIodified Vehicle Program Listing
321
SIJRPO'ITINE SJMKO SI*"XP **,<)« VERSION
r-4, PLANT (SFNSOP-VFhICi F- ACTUATOR'
COMMON V<MI .WI70I .NX.NK.NR.NU.INIT.IFLAG.ROOE .EIAl.Tei.T.IEC
common /ot apf / mar*(poi» locate, insert. null
ntMFNMON xsoot m ,XS Ml .XVOOT 110) .XV( 101 .XAOOTI3I .XAD) «RS(?I • l US(?I .PV(?) ,UVU) .Ra(3) .UA( 1 ) , U (?)
oimension asm.ji .asm.?) .cst?.3) <os<?.?>
1 . av< io. ioi .Rv( to. <•) .cV<?» l 0) .l>vt?.t.)
? .AAn.3i.Atn.ii.ciMOi.DiD.il
0 |MFN< ION I SEN < ?0 I ,IACT(?0> . IvEMt?0>
rou i valence ((S00Ti)>.4<i)).(xvn0T(i).w<«>n.(xtD0T(i>ttt<Ui>.
1 (PS(li.w(i7)).(Pvil).w(l«n.(Q*(l),H(Mii.
? nisi 1 1 ,w(?ai i . div i i » . w ( ?e>» > . tux M » .woo> > .
3 ixs 1 1 1 .wcu m • < x v « i > .worn . (XAMtoritciT.
A 01(11 .W(47> I
INITIAL I7C
If (INtT.NE.O) C.O TO 100 N’l 5 7
C RFAD INPUTS FPOM SfNSOB. VEHICLE . AND ACT JATOR
e i
ofAn|5,->A«| I SrN j /
dfao<s,?<jri ivfh ; *
RE AD ( S . ?RR 1 HOT '
?RR c OPM A T ( ?0 A4 )
o ALL T APF (LOCATE. ISSN. 71 NR I TF ( Q«?RR l JSFN ?AR FORMAT (IX. P0A4I
RE An ( 7 i T.NS«.NSB.nSU,(<AS(T. I).I*1.NSXI.J«1.NSX).
1 I (PS( ( .JI .1*1 .NSXI . J*1 .NSUt .
?< (CS< ( . )) .1*1. NSP> .J*l .NSXI . i( (ns(r.ji .1=1 ,nspi .j*i.nsui
CALL TAPF (LOCATE. I ACT . 7)
WRITE <R.?rtRl I ACT
OrAOITt T.NAX.NAP.NAU. ( (AA( I. I) . I«1 .NAXI . J«| .MAX) .
1 ( (PAM.Jl .I*I.NAX).J*| ,NAU) .
?( (CA(T , Jt .J*) .MAPI .J*l .NAX> ,
1 ( ( nA ( ( . JI . I *! . .AB I . J* 1 .NAIM CALL TAPF (LOCATE. IVF”. 7)
WR I TF ( P . ?t)Q I l VEH
RF An ( 7 ) T.NVX.NVR.MVU. ( (AVI T . I) . 1*1 .NVXt , J*1 ,NVX) .
I ( f RV ( • . JI . I *1 .UV* ) . J=| .NVU) .
?( (CV(T.J) . J*1 .l|VR I , J* 1 «NVX I »
0( (OV ( T , J| . 1=1 ,NVO| . J*| .NVUI NX * "SX ♦ NVX ♦ MAX
NY = « SP » NSU ♦ NVR . NVU ♦ NAP • 'IA J r
C PRINT OUT “ATP 1 X OIIAOP'IPLFS FOP SENSOR. VEHICLE. AND ACTUATOR C
I F ( IF' AO.NF.O) 00 TO IP?
WRITFfR.il?)
11? roPMjT(??H CONTINUOUS OUAORllPl.ESI
C AL L 'POSIAS.NSX.NSX.NSX.NSx.T.fcHjs )
CALL ‘PPSIPS.NSX.NSU.NSX.NSU. i .AHRS I
Figure Cl. Program Listing for Plant Equations
322
t
l
I
FALL -out, <CS .MS3.MSX .MSP.NSx • ’ .fcMCS call ^ops <ns .nsp.ns'i.nso.mS'i. t.ahos CALL "OPS t AV.NVX.NVX .NVX »NV< • ’.fcHAV f ALL .o»S (RV.MVX.NVll.NVX.NVn. T.AMOV C 411 ,oUS<0V.MV9«NV«.NV3.NV« • '.*HCV TALL "°9S (nv.MV9.MVII.NV9.NVP. * .(.MOV TALL '<BOS( AA. MAX. MAX. MAX. MAX. T.4 HA A TALL "OOS (RA.NAX.NAU.NAX.NAil. r .(.MBA TALL "P9S(CA.NA0.NAX.NA0,NAx.t.AHCA TALL <OPS (OA.NAP.NAU.NAO.NAU. T .(.HOA r, 0 TO 104 10? WOITFiq.lll)
111 format <i«h otaital quaopuplFsi
CALL "P9S< AS.NSX.NSX.NSX.NSx . T.fcHFS FALL MOPMnS.NSX.NSll.NSX.NSH.T.fcHO.S CALL mPPS<CS.NS9.NSX.NS9.NSx «t.4HMS i* all "OOF (f)S.NS0.NS'I.NS9.NSu. T .fcHFS CALL "OPS ( AV.NVX.NVX.NVX .NVx . T.fcMFV CALL <0PS(RV.NVX.NVU.NVX .NViI. T * MHC.V CALL "BOMCV. NV9.NVX.NVP.NVx .t.ammv CALL "OPS(nv.NV9.NVI|.NV9.NV'P. T < i*HF V call "opsiaa.nax.nax.nax.nax « i . ahfa
call “PPSlflA.NAX.NAII.NAX.NA'I.T. AMO A CALL UP9S (CA.NA0.NAX.NA9.NAx . T « <*MHA CALL "09S <nAiNA9.NAU.NAO.NAU. t.AMFA 104 CONTp'tIF - OFTlIP"
’100 CONTMliF C
C C0«PIITF OTFFFOFNT t AL STATIONS C
C SENSOO DYNAMICS
00 PA'i 1*1. NSX v ( 1 1 *n.n
no ?oi 1*1. nsu
?01 V(1I*W(I»»«F1T.J)»PIF( )l
no ?r>* i*i. nsx
?0P Wllls><(t».AS(l.J)»*S( n C VFH1CLF DYNAMICS
no ?o?i*1*nvx
T T = 1 .*’S X VMTirO.n no ?o t i=i.nvii
?0O V < T 1 1 *V ( 1 1 1 »BV ( 1 . J l »UW < J 1
no ?0’ J- 1 . nv x
?0? V(TTI*V(in»AV(l..ll*Xtf( I)
C ACTIIATOO ny^AMirS
no ?0i. 1*1. NAY
1 [ = !.«IM»NVX
V( 1 1 1-0.0
no j = i « n a u
?0? VUI)*V(in.RA(l.Ji»liA(Ji
no ?p. jsi.nax
?0A vim = vim ♦ A A I J ..11 * X 0 ( 1 1
c
C COMOUTF O'lTPUT fquaTIOnS C
C SFNS03 OUTPUTS
00 ?* 1*1. MSP II * I ♦ NX Vdll = 0.0 no ? 7 1=1. MS X
?7 Villi : villi . CS(|,.|) « XM PI
no ?a .1=1. nsh
?s vein = vim . nsn. ii ° usi it
C VFH1CLF outputs
! Figure C7. Program Listing for Plant Equations (Continued)
1
A
. . w, .L -^
323
no ?a ? = i * n vo
!! = T « NX * NSB
vun = r.o no ?q 1=1. nv»
2<J Vim . virn . evil, M • *V( I)
00 ?fl Js l ,NVM
?0 V f T I » * V1M1 * 0V1I..M » IIV(I) C ACTUAT0P OUTPUTS oo on 1=1. nab
IT = I ♦ N* ♦ NSB . NVB
VI Til =0.0 00 11 Jrl.NAX
51 V 1 T 1 ) = V 1 1 T > ♦ r* f T • J > * k a ( it .10 10 J*l.N4ll
30 V 1 f I ) . V(TT) » DA ( I • j ) • |j a ( 1 1
r
c rvrrproNNFCTtoN foiiations r
T T =n*.nsb*nvb»nap C SENSOR TNOIITS
V 1 1 1 ♦ 1 1 *BV 1 1 1 V 1 1 1 ♦ 1 > =PV 1 P 1
c vehicle inputs
V ( 1 1 ♦ 1) »PA til V(tI»4)*RA<?»
V( I T . = ) =B A 1 3 )
V(II»AI=IM?I C ACTUATOR INPUT
vin.7i.ijin
c
C PLANT OUTPUTS
c
viii.ai.psm
V 1 1 1 .Q) =RS 1 ?)
" pptupn
END
Figure C7. Program Listing for Plant Equations (Concluded)
324
•**•'• u J.-jaLu^jau£dAs.i
simao ittnf fta“<4
FTA«<T n«.0 r t/roctON
COMMO1' VT<*1 ) .*M70) .N*.NV.NP.N". TNI T • T CL AG .NODE . F ( <t I .70 I . T . IFF roMMO'vsT/snnT (T.i-')«Kn*io*.3HT.io>.iiio.|0».U(7>.NN«<lo>» )NNP( 1 ' I .NNUTIO) ,N‘-'AX. HO, HOM'X.TOF, I 30 f NT . I S I"< < ?0 1 r OMMO-1 /OT APF / maok(?0>. LOCaTF. TNSF°T. NULL niMFN^ION !HF4F(?nt OIMFN^TON LARoH.'.'
DTmfncTON a (?«• PA) .►>' >R.7| « C ( PA) * 0 ( *>• 7)
ptmfn=ton fpi.msipr.pa .fpl'if ( 3«. 7t
DIMCNC TON FT (’«•?«) »r, ' (PA.7) «oT (6.PA) .FT <#>»7)
?..IOFP(?M> . JINOTPPI INTFOco hFLAO INTFr.ro SINF MAP* ( t > = OHAtM.
maok ( ’i = <.h«a*a Lor.4Tr = ohlooa
TNRFOf = <..*|f|Sr
NULL r 4HNIJLL
OF Aft ( O . 77 7) IPJINT
777
FOOWAT ( I?) o-AO ( 77«<) jrr 77R rnoMA’ (I?) MillN:. I
«4*m= »o
MXM = ?«
N3MTf,
N JMs7
FOFFrl .F-.1P H4*iDtO
r r r r
ono
j F | S4md .,r 0 INC JFMFN7 aI QUAOP |P| FF pill PF opjNTrri AND weiTTFN ON ANO L *orL "ATA CAOna FnJ inOPcmcntal ouadPIIPLFS «URT RF INSFPTFO
70 1
FONT I IIF
0FAn<o.7B| i hfi ir,
FTOMA - ( T?l jr (hf. ac,,fi ! i of r i toi i WO | TF TQ.7AR)
7AF FnowA r (HI)
WPTTFtO.Tl tooimt. ift.mFL 40
nJUA* (OH jODpiT It 3.F.X ) OHFL I ‘.HT rONn y T TON = I 3 • f. * 7HHF|_ Af, = j?t
INITTALT7-
TOO
jfihF' ao.fo.Ii on in ?on TNTT r n
OF An f f . TOO ) ( TUF A0( I I . T*1 >?'ll
f noM a r (Thai.)
W 3 I T F <0. ITT) ( T -tFAr ( ! I . 1 = | , ?(n F n 0 M A T ( I < . 30 41. )
rn r
f FOOW NM“' . TFLAO.MnriF
r
nFronF(4,7«A.ioF4nt Qa.nsH”'.tflao.w:)‘)f F00M4 ■ ( A 1 . T T 1 )
O0| TF I Q.7TR) N' IM<, IF( AO.oonf
F 0 0 A T (|A. <SONSIW* = T3. ?X. hu I F L A r,s 1 3 , ?». SwlOOFiJ?)
7AA
7T0
Figure C8. Program Listing for State Modeling Program
325
I
f
[
t
I
1
i
}
h
l
C
c compute r c
IFtMFt AO.EO.ftl M T 7 >4 T«
IFinfi ao.EO.A) r.o to ots
WR|TE<9,77<)t MFL»r.
770 FORMAT <7HHFLA0r I?.14m|S NOT ALLOWF 0 )
stop so
Q7<; 00 TO <901. OOP. 007. 004) NSIMK
O01 TAIL SIMKTS 00 TO AOS 90? CALL A TMKTV 00 TO «0S 007 CALL a tmk T A 00 TO AOS 904 CALL AIHSTC 00 TO AOS 937 CONTINUE
00 TO <A01 .AOA.A07.A04.AOS.AOM NS1M< 001 CALL AJMKS 00 TO AOS AO? CALL ATMAV 0£> TO AftS A03 CALL aJMKa OO TO AOS A04 CALL ATMAC 00 TO AOS AOS CALL SIMKP 00 TO AOS AOS CALL AJMK 00 TO AOS 700 continue
737
730
OpAfU 0,737 ) ISOmax . < IS IMif ( i ) . I-) , I co via « >
F OPM AT ( ? 0 I ft )
WPJTF <9.73ftl ISOMAX. USIMK(I) .1 = ] . Is3v,a«, FORMAT (10X.7MISOMAX=I?,3X.ftMlclMXr?oi7) PFAO (S , 700 I (LAPHIIi ,l = l,?ft)
DFCOOF (4 . 7AA.L ARH) CA .NS (MK , IF| AO.modf
00 7010 !SQ*1. ISOMAX
INIT*0
I f ( I s a MP , pi} . n ) oo Tn 7R7
DFAn< c « 700 ) <IMFAO(n.I«l,?n)
WRITE <0.333) < I me ao < I ) • id ,?n)
703 CONTINUE
WO I TF ( 0 « 73S ) NS 1M< « 1 SO.MOOF , ; c j mk ( I SO )
736 c2u*Ms{iK6HNMHKS,?,?*,4HICOrI?,?*,5HM10r*,?‘,**f->SI^*T?'
AOS CONTINUE IN I T = 1
WOITFfO.SOOSlNX.Nr.NR.NU
500,5 ?X* iMNVrl?, AX, IsM-fR. ?• . 1HNU-I*)
NiNXtnY.NR
IF(MFi AO. EO. 1 100 TO 4ft IFIHFi AO.FO.?) 00 TO 777 OO 101 Jsl.M
101 W ( J | = ", .
no soi /=i .m
W(J>=> .
S??° C,3LLTOST,2Ks"’,?",n*"K,,,,,i’"'M
00 TO AQO
Al? call s imk v
00 TO R96 A17 CALL c INK A 00 TO AOS
Figure 08. Program Listing for state Modeling Program (Continued)
326
#14
Figure C8.
CALL STMKC C,0 TO #9#
SIS CALL simkp C.O TO #06 #16 CALL SIMK #96 CONTINUE WIJI-A.B 00 SOI 1 = 1. N SO! E<t.Ji*V<|l C.O TO #00?
TT3 tWTTMtJT
C
C 7FB0 OUT kOOT.RT.UT.X.II C
no 10 NN<<1<NMAX MX *NNX (NN)
— ' no nr i*i <mt
XOOTI l,NNI=0.0 IS X(J,NNI»0.0
DO 11 NN*1 <NMAX MX zNNP !NN)
DO l? J* 1 <MX
1? - 9T7J<NN>iB.O MXsNNulNN)
DO 13 J«1<MX 13 III ( J<NNI =0.0 11 CONTINUE
00 14 1=1 <NU IT llf!»=n.fl c
C COMPUTE PAPTTALS WR7 STATE DERIVATIVES C
JJ*0
DO SO NN* l.NMAX MXxNNX (NN I
no so j*1<mx
JJxJJ.1 XOOTI I « NN 1*1,
c,0 TO I10n<101?<l0l3,10l4) NSIMK 1011 CALL SIMKTS
no to io?s 101? CALL * 1MKTV C.O TO 10?S
1013 CALL SIMKTA C.O TO 10?S
1014 CALL SIMKTC 10?S CONTINUE
XOOTI l <NN 1=0.
no so t*i.n SO ni.jntxin
c
C COMPUTE PADTIALS MBT INTERNAL OUTPUTS
c
no inn nn=unmax MXeNNP INN! no ion j* i .mx JJ=JJ<]
R I I J.-INI = 1 .
no TO I POO I <?00?<?003t?004) nstm# ?001 CALL SIMKTS
no to ?n?s ?00? CALL simktv no to ?o?s ?003 CALL s I MK T A no to ?o?s ?004 CALL s|MKTC
Program Listing for State Modeling Program (Continued)
327
, IPMHNI llipjlll
■ri
PO/’n |
FONT | 'll* |
ot ( |
|
nn i o > t = i .n |
|
100 |
F< J. J II =V 1 I | |
r
C C^MDiirr P-JOTIALS W^T U'TF^Al |MP 'TS
r
nn jc; NNxi,N^a<
*» « = ^N ' ( V |
no i*; ■ , i .««
)J = J.l.l IIIIJ.'lMItl,
00 TOi 0«0| oFim*
?noi CALI f.'MiOS no TO W11;
?00? fall . ;mkTV O0 Tn 001F
O^OO fall ‘ I mk T A
00 TO »<I1H 0004 FALL MMKTC
on TO
?" os font i iif
'If ( = d.
no if T * 1 «fi ISO Fit. j n.vi n r
f r0MOi|Tf P.DTI4LS -QT staTcl
r
no on nn= i .mmak
««=NN' ( NN I
no ?0-' i=l • ma
J l*JJ. 1
« (J.M-.|-| .
no \*stmk
?04| FALL FJHKTS no m ?04S
0040 FALL :»M«TV
n n to ?'4S •141 FALL ofMKTA F,0 TO O.OaS Tout, FALL ; T"AKTF OliiS FONTI'IIF
« (,i.»i"i =n.
n » »<>•• r*!.N
?no r ( i . i ii *v ( i i F
F fomomTf PaoTIAlS POT FuTFwmal (mO"TS
F
no of J=| ,NM j i= ij. I 'M.n = ■ .
F. o TO (Ofist .OOSP.OnSO.OLSfc) NFJMK ?(ifi tall «?**«rs F,n tfi offs
on®? fall Ffpicrv
00 TFI ooff
?3FT fail IHAT*
on TO OnFF
?0S4 fall ' T TF
OQFS FONTT-'IIF "I 11= •. no ?S ■ 1 = 1. -I ?S0 F ( j, j i| =v ( T t 40 FONTT 'OF on 4i i=|. m 41 P( Jl=n.n
Figure C8. Program Listing for State Modeling Program (Continued)
328
03 4? J*|.»
W< JIM .4
CALL “St** nli>«4.o
00 4? f = i ,N
4? r(i.ji«v(ti «oo? crwTpnir c
f FORM A.P.'-.O MATOIfFS
r
N\/=NX.NV
JF(IPo|NT.FO.et 00 TO 15 r ALL <‘PB4<F,MAKN.M*xM.N.M.T,4wF I IS rONT!“'lF
00 SI I«1.NV 00 5? 1=1. NV 5? Fn.j>»-r(i.jt 51 F<I.Ii«F(t.!).l.
CAU. TOJNWei l SOL *10*01 .NV.-M.r-,MAXN.<DU'«.DETI
I P*NV • 1
IF«NV*NP
JP*lfl
JF«M
00 53 I « I H , IF 00 53 .l«JR«JF no 53 *=1.NV
53 r(i.ji«F(I.jl«F(I.*>*F(K,j> no 53i l»l. IF
no 53' J*l. JF
IF(*flc(F(I..n l .LF.rPSFI Fd.Ji * n.O 530 CONTI mi IF
iFdPoiNT.ro. oi r.o to 54
WOITFMS.500T)
5007 F3Rm»t( /7X,|*H <.imiL»TlON ••*TBIXX)
CALL mPRSIF , M AX N • M a XH , N « M , T « 4mF )
54 CONTINUE Jl*NV.l J7.NV.MX J1»J1 .NX
Jfc.J7.NU T I *NV ♦ 1 IP=NV*N» no 400 1 1=1 .NX
no so 'i 1 j=ji . j->
jj=j- n*i
4001 *<i.j iixrii.j)
no 40''? 1*1 ,N*
00 40r? J=J3tJ4 JJ*J- IT* I
600? P d * J II *F d • II no 4043 1 = 11 ,1? iT*i-d*i no 40 n t j=ji , j? jj=j- ii*l
4003 CdI*.IJt=Fd*Jl 00 40,'4 1 = 11,17 TT«I-I1»I
00 404fc J=J3.J4
JJ=J- 13*1
6004 Odl* IJ*=F(|.JI C
C OUTPUT n.p.c.n >*ATPlrFS C
IF( d^AMP.FQ.O) .ANn. (mFLAG.fO.I ) IGO T3 780 IFIHFi XG.rO. 1 ) GO TO 4006 IF d Ft AG . NF . 0.1 GO TO 400*
Figure C8. Program Listing for State Modeling Program (Continued)
i
MtOS *0|TF <N.?I
? foomat < 7« . iskfont t minim nnnr/>
CALL AMf (It
FALL *0WSIA.N»*<%N«m,N«.N«.T.i.-A \
C*IL -OPS (R.N*M.NIIM.N« »MU.T.« -P t
C»LL "PBS< F.NBN.NiMtNB.Nll.T.'.HF t CALL •<P»S(n«NBM,NU“.N;»,NU.T.*..n t
f.o to ?po ftons font i ' iif
KPITF (O. SI
S F0H»»*T(7*.t?t40TOTTAL MnOE/t IF(HFl AF..FO.I I tfBJTFlp.M A ro»M4M1?M tNC°Ft*FNTAL t
IFlHFi AF..NF.I »r*LL NAMFIll FALL *PP5IA.NXM.NXM.N7.N*.T,<..>F I
CALL iOBS(B.NXM,NUM.N«.NU.T.A-ir. I
FALL ‘'BPS! F.N»H.N«M«NO.NX.T.'.HH >
FALL 'PBSin.NOM.NUM.NB.NU.T.fcuF t
780 FONTJ'iliE
IF ( ( HANP.FO.Ol .ANn. (MFLAO.rO. 1 ) » 00 TO 78?
C
C POSITION T4BF AT f‘T> OF L 'ST PFFO-I)
C W»ITF LAflFL ON TAPF F
CAt L TAPF (INSFPT, IHFAO. 7)
ttPITF (711. Mi.NO.MIJ. ( <A< I. II . [si .NXf. J«l.N*t . 1 ( (O ( I ..It . I*| .NX) • 1=1 .Nil) .
? (( F ( I .J) ,I=|.NP> . 1=1 .NX) <
3 ( (O ( I.J) ,I*| .NX) . in.NU)
CALL r APF ( tNSFDT .'*«P« . 7|
78? FONT]- ItF
IF(hFi AO.NF.1IO0 TO BOO IF ( ISo.nf.I 1 00 TO 700?
r
r INJ TT AL 1 7r FPtUS.OPLUS C
TOTT = ■.
OO 70^0 I * j ,Nt 00 70«0 J*t.NX FPUIS ( I . Jt *0.0 IFIt.cO.JtFPUtSU. 11*1. 0 7000 OONTI'HF
no 70-| 1*1. N« no 7fl q | J * 1 .Nit 7Q0 | 00| '1*1 1 . J t =0 . 0
700? FONT | ''I IF C
r IIPOATF FPi HS.OPLIIS
r
IF ( IS»«* ( ISO) .FO. 1 I TOTT *TOTT ♦! no Tflqs 1*1. N« no 70"S J* 1 .N*
f< i,.n*n.o
PO 70 i* K* I .NX
7005 FH.JI* F( I , j, .A ( I ,K)»FP| US(K.J)
no 70"A 1=1, N*
no 70"7. J=1,N»
7000 F?LUS'I.JI= F ( 1 . I) no 70"7 1=1, N* no 70 0 7 J= 1 .Nil
F(i.ji=n.o
no 70 ’7 K=|.N»
7007 F(i.J)= F ( I . J> ,A( I .K)»0P| US(K.J)
no 7010 |=1. NK no 70^0 J=1.M'(
7000 oBLWSd.jis f ( i , ,ii »n ( i , ji
Program Listing for State Modeling Program (Continued)
330
m
( *r,a«P.fO.O>r,o Tn 7010 ITT f R. 7<.M ISO. ISJMK ( I SOI 740 FORMAT (S».4HlS0*l/>. ’X.hHlSIXKr I?»
TALL iRRSiFPLnl.NXM.NXM.NX.NX .TOTT.AHrPLS!
WO | TF (0.74M ISO. HIM* < HOI FALL 'OSS < FiPL 'IS .NXM .N'lM .N« .Nil . TOT T . 4M0PLSI 7010 FONTI'iIIF
: PTnilNRANC* CnEC* 0.' STATE VARJAoLcS
S F*TBAfT1NF. REDUNDANT STATfS
jn*n J ! *0
no 4 3^ j*i .na no Ala i*i.nx
TF ( ARS ( FPLHS 1 1 « J) ) . (it . F°5F ( no TO 4?0
410 FONT 1 \n IF
no 41S 1*1 .tgo
tf (arc ic ( i . jii .r.T.FPSFtr.o Tn -.on
411 FONTImIIF
jn*.m.i
jnfP( in i « j F,0 TO 410 4?0 FONT T‘ HF •IT* Jt ♦ I jtNni iti *J 430 CONTl'lUF
FO°M I NO TmF RFOUCrn A.P.C AND 0 MaTPICFS NX*J!
no 47" 1*1 «N*
1 1« 1 1 no 47o J*1,NX JJtJNlIJ!
470 FT ( I • lIsFPLUSdl.JJI no 4R0 I* 1 .NX II»JlMn<D no 4A0 J*1 ,NU 410 RT(T, n*RPUTSTIl.JT nn 49"i i*i. nr no 490 1*1 .NX JJ*JlMO< JI 490 HTU. II *C ( T . JJ1
no soo i*i. nr
no ion j*i.nu
100 ET(I. M*0< I.Jt T*mr WR I TF (9.7)
’ FORMA T ( 34H QUADRUPLE OVER ONE PROGRAM PERIOD! CALL mPPS I FT .NXM.HXM.NX.Nk.T. 4HF )
0 ALL “PBS (F.T » WXN.’VUM • MX . NU « T .4Hfi !
CALL mpRS<HT.NBN.NxM.NB.NX.T.4MH !
CALL MPB5 (ET .NOH.NI/M.NR.NU. T • 4HE >
at sequfnff end store total r r, h r oh mao tape
CALL tape rTNSETAT. LXWH, 7!
RRITEI7I T.NX.NR.NU.KFKl, J).!*1,NX),J.1,NI). 7TrGT(T,J),I*l,NA!.J*J.NU! .
?( (HTI I .JI .I*| ,NR) . J«1 .NX) .
3 ( (ET ( T , J) . I *1 »NRI . J*1 «NU)
CALL TAPE (INSERT. MARK. 7 |
no T0 9<nr
END
Figure C8. Program Listing for State Modeling Program (Conrludnd)
SUBROUTINE HM««
C
C HS1HK S6"0 VERSION
c
COMMON V (41 1 ,W< 70) .NX. NY, NR. NH. INI T . I EL AG.MOOE ,E 1 4 1 , 70 ) , TT » IFC COMMON/S3/XOOTI3* 10) «X 13,10) .ol ( 3. 10). DI O. 10), 00(7). NNX (10). INNR(ln) .NNIHlOi .NMax. ISO. |SQMax,TPS.I*RINT.ISImki?0> common /OTAPE / MAPKIRfl). LOCATE, INSERT. NOLL DIMENSION ICON (?0 ) • IPL i ?0)
DIMENSION XPPLUSI 141 «XCPLllSl7l .XHPLUSIl) *XM0LUSI1>
I ,XP (1 At .XCIT) ,XH| 1 ) ,XM I | > t . RP I ? I . RC I 1 I . PM ( 1 ) . PM ( 1 ) ,R m l.UP (?) .llC 111 «l)H( 1 ) ,UM 1 1 ) »U< 3)
DIMENSION EPIlA.IAl .DPI !&.?> ,*P|?.16I .EB<»«?) l.rC(7.7).fiCI7»T).Mr(l.T),EC(1.3)
?.NSTA->TI?0> .TI101
EQUIVALENCE (XBBLUSII 1 .MU)), IXCRLUSI I I .Ml 171 >. IXMPLUSU) .M(?4I I . I l XMPlUS ( I ) .MIPS | ) , <uP( 1 ) .MI2AI i • (RC( I ) »M(2S) I . (PHI 1 ) »MI2<)I ) .
2IRMI1 i .Ml 30) 1 . (UP » 1 ) .Mill) ) .I'tCII ) .M(73) ) . (OHII ) .M(36) ) .
3(tiM( 1 i .Ml 37 1 ) . (*P < ) ) .Ml 3A| ) . («C 1 1 1 .Ml 34) ) . UN1 1 ) .Ml 61 1 ) •
4(XM|| < »MIG2> ) . UK I ) .Ml All )
IE ( INTT.NE.O) DO Tf) 10 C
C INITIAL!?1- C
NKM.l NOH* 1 NllM*l NXM»1 NRM»1 NH“ = 1 NXP*|*
NOPa?
NilPi?
NXCrT IRC* 1 MtjCz3 NR = 3 N 1*3
N»«N*'>.NXC*NXm»n*n
NY*NRh.NRC*NPh.NOM.mUP.NUC*nUm»NUm
NSTAOt (11*0
N START (?) aNSTAOT ( 1 > »NxP NSTAPT (3i«nSTaoT(?i »N»C NSTAOt (41 aNSTAOT ( 3t «Nxh NS Tart (St *NSTA°I (4 I .N«M NS T APT (4 1 sNSTART (S) .Nod NS TART ( 71 *NST A°T (4| «NOC NS T ART (H) zNSTART ( 7 I .NRM NSTART (Qi zNSTART ( « 1 .NJ»
NSTAOT ( | 0) zNSTART (01 .ni|R NST AP' ( ! I ) zNSTART ( | 0 I .»"IC
NSTAOT ( I?l zNSTART (IT) .NI|M NSTAOT ' ’ 31 =NSTAOT ( ( R I .MUM C
C OTTRirvr ^ONTROLLr J DAt5
r
If ( I S'-. NS. 1 I r,0 Tf) Rfl 1 ?0S CONTI 'lir
OCADft.m?) 1 1 CON IT). | a 1.?')
Figure C9. Program Listing of Subroutine HSIMK
CALL ' APF (| 1'(|| . I 'r\- , *(
N? 1 Tr 1 a. 1 0>~ i i t rev (I, ,
BC*r* ( M TC *N»C •*i-T . •* If . I ( FC | I . 1 1 . I r | a M« r I • 1*1 . N«ct •
I Kr.r< i . ji .|ri .-uri . <s| .•mn .
?( («r( i. n . |r|."->r i . i- 1 . *<r i .
H«Ff 1 1. JI . | = | .-isn . 1=1 ,N(in
CALL “OPS(ro.N»C.N»C.Nxr«NXC*7C.‘.“ rc»
TALL •|0k'S(r,r.N«C«‘JiiC.N»r*Ni)c*’C.4H r,z I ran. >'0>»«<Hr.Mur.tj*f .Wi>uc.!r.<.K «r>
TALL 'BUS 1 rr ,N Pf .N*i0 «"|UC »NIIC • ' C . <.* (■•'•I
?0I CONTI'iMF
- PCTQfFVf- ILA'/T TATA
lrt IS ..Nr. It r,') TO ] fi 1 IDS CONTI*'* *F
OfAO (C.IrtTl (lOl.ip* I = 1,->(1)
)C7 rio«AT (?(1A<.|
C4U * ABC (L^r ATT . TP| . 7,
WPJTF (Q.10M ( I PL I I > . I = 1 . ?*M 0* F OWN A T I 1 > . 1
orAom tP.*;«t>.*:jp.. ,,o. < (p0( 1, n . 1 r | .sj»di . j.| ,v<0| .
I ( (<>P ( I • l>.| = |.'l»P|, 1=| • NUP 1 •
?( (MO( 1 . n . 1 = 1 .*.OP| . j=| .N*J) ,
K (FP( t , j| . | i| .**OPl . 1= I , KH ID 1 CALL ‘Bus ( cp , \i x p , 'j* p ,kjxp a , 1 u , f j)
CALL "OPS (c,p,n»3«N iP.N*P.Nl|o. TP.fcw p^l C*t L “BPS ( wD .N-JO. N»P .NPO.N«D . TP.faU HS)
TAIL “OPS I CO .\|PO . 'll fP . K|PP , >1110 . *D , f}|
Cl CONT J * * IF
SFT TIMIN'. TAHir
F°ST=. 1F-P||
INSFOT T I < I NO T A«Lr
PFTI1Q-
!> CONTI MiF
INITIAL I7r ALL STatfs
00 II 1=1. NF 11=1 *'1401 1 1 7 1
I VI 1 1= ■( M I
Of) TO ( I . p. 0 , 1 I S T “< t 1 so I 1 CONT I • 11 IF
■ iiboatf d( Mir siAir
TT = TC
no iflp i=i.m<d
11 = 1. CTAUT I I 1
vnii-i'.
no 1 n i j=i ..mo
loo v 1 1 1 1 -v ( 1 1 1 .sni 1 . |, oho , ,,
cc 1 n 5 |=| . *J X J
1C? VI I 1 ) -1/ I I I ) ,FJI I , I > 3 HU I II
c.O TO S
CONT I * *1 IF"
UPOATf m TPOLLFo FTATr TT = TC
00 pop I = | .mr
Figure C9.
Program Listing of Subroutine HSIMK (Continued)
333
U*|«-FTANT( ?i VI T T > -r .
00 ?0 > J*|.NUr
?0’ v(tTisV(Ui»<irM> iHMiri.M
no ?p-> j*|.nxc
?«? v m n -v 1 1 1 * *rr ( i . j»»xr(.n r, o to f 1 rONTpMIF r
r tODATr HO' n STATF C
tt=fp ;t
no in’ ►mu II*I« 'FT APT I II
10? 1/(111 si l« (II
AO TO f
a roMTi'-iir
r
f U»n*TF MF'.flPT STATF TTsFPfT
m id’ !*l,NXM It »!»•!«'. ART( 4 1 to? vdiisOMtn s rONTlMUf
r
r COMPUTE ?' ANT OUTPUT no F0’ I«1.NPP 1 1 * I ♦’ ‘FT ART ( F I VtlH’0.0
00 FO’ J*l.NXP
F01 V(tllsl/(III»MP(I.J|»XP(J>
no fo’ j=i,miip
FO? V(tn*V(in»FCM!,J»»UO(J> C CONPUTF COVTROLLfR OUTPUT 00 AO’ 1*1, N»C t !*!♦ 1ST ART ( A I VdllsO.O 00 AO’ J*1.NXC
A03 Vd|)*Vd!>*HCd.Jl*XC(J> DO 60’ Jsl.NUr
AO? V(IIIsVIIIl.FC(I«J»«UC(Jl
c
C CONPUTF MOl.0 OUTPUT
c
00 70? 1*1. NQm It*! ."START ( 7)
70? VdllsXHd)
C
C CONPUTF MCMORT OUTPUT no «0? 1*1. NPM II«I*mSTART( Bl AO? VIIIItlMin
c
C INTFRCONMfCTION fouattons
c
C CONPUTF P| ANT INPUT C
n*NSTABT( r>
vcTT^iiNPmn" ■
V ( n . ’ I *U ( 3 )
c
C CONPUTF CONTROLLER INPUT C
1 1 *NST ART (10)
Ydiurajm
V ( 1 1 • ’ ) *RP ( ?>
Figure C9. Program Listing of Subroutine HSIMK (Continued)
334
<ao> adiitKiT#A,iw>nifi*,«rt iiadii
dtikxMtiittfMit
V( ! I. I|:WP| | t
r
r rO«oi|Tf - >i n I »jt>i | T
r
I | =*^ r ftuT till
V f 1 1 ♦ I ) jWm ( 1 ) . i ( ? |
I r (MnoF ,fij. n v 1 1 1 ♦ n 3 up)
r
C riMBIITF Mi UflBV IMBlIT r
T T *MST ART ( 1 ? >
V I 11*11 «BC( | I f
C CImbiitf <;v<;tfm niiTonT C
ItiNSTAPTini
vu!»n«RP(n
V<
vii r .it »»c 1 1 1
OFTuPm
FND
Figure C9. Program Listing of Subroutine HSIMK (Concluded)
335